Math is pretty inaccessible to most people, I’ve spent a fair bit of time studying it, but I’m not foolish enough to say I really know any. I sincerely doubt that, even if I was given an infinite amount of time, I could have ever discovered the simplest proofs one learns through in high school on my own.
This graph is an attempt to describe theorems that can be conveyed in short natural language prose that most (English speaking) people understand. You would think it would be hard to whittle things down to such a small list, but some googling suggests that many would come up with something similar. One easy way to find these is to look for lists of the “top 100 theorems” or theorems that people have tried to make expository books for.
Outside In – Finding a way to get at the sweet
purple insides of a yellow ball.
There are other short, interesting theorems not in this graph but they require things extra qualifying statements or background knowledge (e.g. the abc conjecture or the Robbins conjecture). All of the theorems in the graph can be explained fairly easily; would love to hear any suggestions you might have of other theorems that should be up here.
Just because we can state a theorem (even one that sounds simple), doesn’t mean we know it is true. Truth is asserted by the existence of a deductive set of steps that most experts agree is correct i.e. a proof. Most interesting theorems require many years of exceptionally capable people’s time to prove that they are true. A great deal of interesting mathematical machinery is invariably constructed as a result of (dis)proving a long standing theorem. It is interesting that there are a few simple ideas, which most of us take for granted (like the Jordan curve theorem) which took some smart people a long time to actually prove as true. On the other side, I think that Cantor diagonalization is one of the most accessible mathematical ideas there is which leads to a profound result (transfinite numbers). This is perhaps no surprise, given how often it is explained in layman’s books.
Some interesting ideas come from this graph about the set of all interesting theorems:
There are relatively few interesting mathematical theorems that can be conveyed adequately in short natural language prose.
There does seem to be a trend of more counterintuitive ideas having more difficult proofs.
There are plenty of exceptions to this trend, e.g. the proof for the four color problem was generated by a computer; no one thinks a short concise proof exists (but who knows, maybe there is).
It seems that few people are interested in proofs of things that are only just past intuition and are difficult to prove (this is perhaps a tautology).
None of those parties I’ve referred to is giving you an honest story. A careful set of explanations will show you that GMO crops (as they are cultivated today) are no riskier than any other produce and are generally not consumed directly by humans. That’s why the whole “GMO free” label is a bit of a swindle; it’s a marketing trick designed to prey upon the ignorance of how GMOs are grown and used.
Which crops are genetically modified?
Very few crop species, but we grow large quantities of the crops that are genetically modified. GMO soy, corn, cotton and canola are grown at a massive scale but nothing you get from the fresh produce section in your grocery store has been modified. Almost all the soy, corn and cotton we grow is either Monsanto’s “herbicide tolerant” or “insect tolerant” type. There are a few others (like Hawaiian papaya) but most of these are grown at much smaller scales and the reasons those are modified are very different from the major GMO crops.
Why are crops genetically modified?
Agriculture is a resource intensive practice, but not all agriculture products go into human mouths. Crops have been genetically modified to reduce the amount of input resources (including energy, water, pesticides etc.) required to grow them (i.e. to lower costs). Most of this crop volume is itself an input into products like beef, fuel and textiles.
The United States grows a lot of soy and corn. Only about 20% of the soy and 12% of the corn is actually eaten by people and a lot of it is in the form of fairly non-essential foods (corn chips, salad dressing etc.). We grow this stuff because humans like meat (which isn’t a very energy efficient way to grow food) and because governments have policies about fuel production. A lot of people think growing corn for fuel is a really stupid idea.
The principle genetic modifications in GM crops come from inserting genes from common soil bacteria, genes that we do not believe pose a risk to human health. Insecticide resistance is introduced from inserting a inspect-specific toxin gene from Bacillus thuringiensis, which we discovered after using crystals of the bacterial spores as a safe and effective insecticide. Herbicide resistance is introduced by using a gene from Agrobacterium tumefaciens. This specific bacterial species contains a gene which allows plants to break down the herbicide glyphosate, and has machinery to insert this DNA into plant cells.
Glyphosate is widely acknowledged as a remarkably safe and effective weed killer, and health concerns are centered on farm workers (who are exposed to many many hazards), not consumers. Both of these common soil bacteria may well be part of the humanmicrobiome. Nothing is ever completely certain in science, but it is very hard to imagine how the existing products from GMO crops could be harmful to human health, and if they were, we would expect the same of many natural and traditional agricultural products.
Am I eating any GMO crops?
Probably not directly, as the corn we eat in the summer isn’t the same cultivar grown for animal feed. It’s possible the edamame in your local grocery store could be GMO.
Since we grow so much corn and soy, it is really tempting to take some of the refined products (starches, oil etc.), throw some sugar and salt on it and put it in some brightly colored packaging with a celebrity endorsement. Processed food products (think anything that you buy in a box: chips, cereal bars, etc.) are where these crops get into our mouths in the most significant way. The products from GMO crops that are used in processed food products aren’t necessarily bad, but they just tend to be marketed in products that you could probably live without.
Are GMO crops bad for human health?
Any diet related questions have to be answered very carefully, since diet science is fraught with a notorious amount of uncertainty. The common sense answer is “no”, GMO crops are not dangerous to human health and we have no reason why we would predict they would be.
One could make a crop with a genetic modification that is quite poisonous, but GMO crops are not about poisoning people. I wish I could say that it is an absurd notion that a society would plant a crop at a large scale to poison people, but we do have tobacco and alcohol which humans enjoy for their stimulant effects at the cost of significant health risk.
You are almost certainly already exposed to the products of the transgenic genes in GM crops (as they come from common soil bacteria) and we have a good understanding of how those genes work and we cannot predict why they would be harmful. GMO crops have also been carefully tested in animals and we don’t see any significant effects. There are plenty of natural things to be worried about when it comes to agriculture (salmonella, listeria, alphatoxin), heck, the natural trypsin inhibitors in raw soybeans will make you quite sick if you eat them without treating them with “wet heat”. It is hard to rationalize a food system that outright poisons its consumers at large scale.
The “golden rice” project is a cause célèbre of GMO foods. The consortium behind the project has managed to insert two genes into rice that enable the production of β-carotene, giving it a golden hue. The aim of the project is to address vitamin A deficiency in developing countries with lacking diets. A really interesting fact about golden rice is that after inserting two of the expected three genes we expected it to be red, since they only provided the pathway to lycopene (the red in tomatoes) and not all the way to β-carotene. It turned out that rice already expressed the (previously undiscovered) third gene and could produce β-carotene with just the first two genes, a testament to Nature’s messy ball of partially maintained genetic spaghetti code. The project has made some remarkable improvements in vitamin A yield and its aims are commendable but it remains to be seen how effective it will be in addressing vitamin A deficiency in the real world. It is hard to argue against the idea that golden rice could be an effective solution and worth trying.
Can GMOs help eliminate world hunger?
We have plenty of food security in the “developed world” and animal agriculture demands we spend far more in terms of resources (energy and water) on food than we need to.
One might think that lowering production costs could help address hunger in developing countries but it is important to remember that world hunger has relatively little to do with production capacity. World hunger has much more to do with things like distribution policies, storage technology, colonial history and water security. Oxfam has a good introduction.
GMOs do not do much in terms of solving hunger; human societies use hunger as a method to ensure certain people are conveniently prevented from pursuing many of the benefits we enjoy in the developed world. Perhaps if we can introduce a gene into our crops that makes the people who eat it more likely to share resources there is a chance for more direct impact.
What will having your genome sequenced reveal about you? Long life? Luck? Impending doom? I think that many of us are under the impression that personal genome sequencing is now both accessible to the average Joe (kinda sorta true) and provides accurate predictions to effectively guide treatment for most patients (ummm…errr….).
My hesitation in responding to the second claim isn’t to say that believing in the promise of personal genomics is anything other than an eminently reasonable position. Think back to the year 2000 when Craig Venter of Celera Genomics (a commercial company) and Francis Collins of the NIH (government funded) effort sat down over a choreographed beer and pizza summit to work out their differences and jointly announce the completion of the human genome sequence. The year 2000 seemed to be as good a year as any to make the announcement, even though there were still several years of postdoctoral blood to burn to get past (a very blurry) finish line.
During this time, the popular press was eagerly imprinting grand visions upon the public’s collective mind. There are many quotes, but perhaps the narrative is captured best by Bill Clinton’s announcement:
It will revolutionize the diagnosis, prevention and treatment of most, if not all, human diseases. In coming years, doctors increasingly will be able to cure diseases like Alzheimer’s, Parkinson’s, diabetes and cancer by attacking their genetic roots. Just to offer one example, patients with some forms of leukemia and breast cancer already are being treated in clinical trials with sophisticated new drugs that precisely target the faulty genes and cancer cells, with little or no risk to healthy cells. In fact, it is now conceivable that our children’s children will know the term cancer only as a constellation of stars.
Questioning the value of the knowing one’s genome sequence may seem silly to some, after all more information can’t do us any harm, can it? Information on its own certainly doesn’t do any harm, but acting on information in face of risks and uncertainties can certainly do more harm than good. Consider recent calls to reduce mammogram screenings, pelvic exams and colonoscopies in certain populations.
Now we need to be clear here, a small set of diseases do seem to be related to single mutations; these include cystic fibrosis, Huntington’s and sickle cell anemia. While we don’t need genome sequencing to diagnose these conditions (they were genetically characterized long before the human genome sequence was complete), understanding the underlying biology of these diseases is clearly worthwhile. The problem with many other diseases is that we are having a hard time finding useful treatments, and the “one gene one disease” narrative doesn’t seem to apply to a lot of things (especially cancer). Even the idea that we have only one genome is being looked at critically these days.
So is human genome sequencing useful? There are a lot of “rah-rah” articles that have a sort of apologist tone (things are “finally” changing!), but lets be a little more critical and split the question up into a few smaller ones.
Is human genome sequencing useful for biological research?
Yes, with complete certainty. The fact that we can almost effortlessly look up a gene sequence in a given organism and run all sorts of interesting comparisons has enabled a cornucopia of wonderful (biological, computer and statistical) science that has fundamentally expanded our understanding of what life is. It is important to remember though, great science alone is not sufficient for finding cures to diseases.
Has the human genome sequence led to the cure of any diseases?
This is a little difficult to answer. Yes, knowing the gene sequence of proteins involved in disease pathways has been utilized to deliver effective treatments to the market (but I am almost sure that every drug target gene sequence was found before the human genome project). Using gene sequences of individual patients is a different story.
Does personal genome sequencing offer any benefit to patients in terms of predicting or treating disease?
Not for healthy people, since the tradeoff between the clinical benefits and risks of sequencing don’t make sense in terms of current practice. The article in the previous link does go on to demonstrate effective applications of clinical sequencing for appropriate patients and suggests a very careful approach including pre-sequencing counseling so that patients can be braced for trying to understand any test results they could receive (perhaps something like this comic could be a part of a standard kit). Perhaps the most common instance of clinical sequencing today is the partial sequencing of tumor samples for some cancers to identify subtypes susceptible to particular treatments.
Will personal genome sequencing ever be useful to patients?
No one knows yet. I don’t think it would surprise anyone if partial sequencing did play a significant role for the treatment of some diseases, but you have to remember how difficult a time we have when it comes to discovering cures for diseases. No one really knows how to do it; there are some of us who think an army of monkeys might be just as good at it as a building full of people with chemistry degrees.
Genome sequencing cheerleaders like Eric Lander have remained staunch in supporting the imminent value of personalized gene sequencing. I think few people would have a problem with the cheerleading if only proponents would care enough to provide the public that funds them with a little more respect and realism about what we think sequencing can and will be able to do. But, then again, measured words aren’t really a good way to go about getting government funding.
Most students are introduced to the “golden mean” (approximated by the decimal number 1.618033988 and denoted as the Greek letter phi (Φ)”) as what the ancient Greeks considered to be the “perfect ratio” with the implicit suggestion if you don’t think that rectangles with a Φ:1 aspect ratio are the best damn rectangles you’ve ever seen in your life, there is
something wrong with you. Now, these golden aspect ratios make for perfectly respectable rectangles, but it was never obvious to me what made them that special. I thought it is a little odd that humanity would have a consensus on such a subjective notion as beauty. A google survey finds that this idea is alive and well in children’seducation (far more so than the tongue map from we looked at in an earlier comic). Perhaps this notion deserves some closer examination.
The number Φ can be defined in a few ways, one being the limiting ratio of successive terms in the fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34…). It has inspired generations of Φ-o-philes, the first of whom I can find seems to be Adolf
Zeising, who wrote the following in 1854 (from the wikipedia link):
“…the universal law in which is contained the ground-principle of all formative striving for beauty and completeness in the realms of both nature and art, and which permeates, as a paramount spiritual ideal, all structures, forms and proportions, whether cosmic or individual, organic or inorganic, acoustic or optical; which finds its fullest realization, however, in the human form.”
We can examine a crystal structure of DNA and find that the ratio of the helix diameter to length of one turn is about 34 Å to 20 Å (1.7), that is not Φ. A universe in which the helix length to diameter ratio was Φ (or even, say, 1.62) would likely have bizzaro physics compared to ours. Nautilus shells have log-spiral coefficients of about 1.33 (not even close to Φ). Given the variance in human anatomy, there is no reason to believe that various anatomical ratios approach Φ in any meaningful way. People superimpose all sorts of thick lined golden rectangles over Parthenon pictures and blueprints, but not only are the lines positioned rather arbitrarily, there is no evidence that the builders (who worked a century before Euclid was born) were even aware of Φ. You can find hucksters on the internet trying to sell you a slice of Φ for all sorts of things including guitars , stock market trading strategies and beauty products.
What drives people to claim structures fit the golden ratio for any number in the neighborhood of 1.6 and do so in flat denial of clear facts? Perhaps it is merely the old trick of imbuing a work with authority via complicated diagrams and mathematical symbols (“Geometry! Greek letters! Math formulas! See?”). After all, we do see numbers like π (“pi” , defined as the ratio of a circle’s circumference to its diameter, approximately 3.14159…) in all sorts of important formulae . It isn’t so much that we find π in a lot of places, but more that we find circles and spheres a useful way to model physical things. Tree trunk cross sections are certainly not perfectly circular, but if we pretend they are (along with some other information), we can easily make a reasonable guess about how much a wood might be in a forest.
This isn’t to say that Φ is never involved in parameterizing models of the physical world (it is, after all, the solution to a simple quadratic equation and is easily constructed geometrically), but its use isn’t any more significant than other dimensionless numbers like the square root of two (1.4121…) or Euler’s number (2.7182…). This doesn’t mean that all scientific publications that invoke Φ are valid though, some of them are considered to be pretty stupid .
An example where there does seem to be something interesting happening is sunflower seed patterns, a favorite hypothesis of Alan Turing (the father of modern computer science). Apparently the first time anyone has ever decided to look at this hypothesis critically is in a recent project , asking students to grow and send in sunflowers so that their spirals can be counted (look at the site to see how this is done). I’m a little concerned by their sampling method bias as well as the fact that they haven’t actually released their dataset yet, but it does seem like about 80% of sunflowers do have spiral counts of fibonacci numbers (I presume the spiral counts have to be 21, 34 or 55). Perhaps there is something special going on here, perhaps there isn’t; the fact is we still don’t have a good understanding of the bio-molecular process of phyllotaxis and there could be completely ordinary reasons for why we observe these patterns in sunflowers. Recall that there are many plants that exhibit growth patterns that don’t seem to have anything to do with golden mean, so why does this allegedly universal law only show up in things like sunflowers and pinecones and not seaweed, grape bunches or maple leaves?
The most common references to Φ are the supposedly “hidden” references in the compositions of classical paintings. You may not be aware, but there is an exciting classical realist painting movement that is gaining momentum across the world today; there are more ateliers that offer instruction than ever before. Many great artists pumping out stunning and inspiring work , of which I am a big fan. Perhaps this is why I’m especially bothered when I find skilled and otherwise knowledgable people parrot the same old fibs about Φ.
A well known book amongst artists is Harold Speed’s “The Practice Of Science And Drawing” written in 1913. The very last pages in the print that I own are devoted to pointing out particular length ratios between conspicuously chosen points in classical paintings. No doubt the particular points he’s chosen lead to Φ-like ratios, but can we really believe if we selected the same sort of points in those paintings that we wouldn’t have found other ratios just as frequently? Couldn’t those ratios be just as close to say, π/2 or 5/π? The widely respected book “Classical Drawing Atelier” spends some time trying to explain the origin and importance of the golden mean while committing same factual errors that have already been cited here. The golden mean is invariably mentioned in (terrible) “how to” websites about drawing and painting, which no one should ever use for anything (especially learning how to draw and paint).
Many of us lament the poor quality of childhood art “education”, as despite taking art classes every year, few of us learned to draw past an elementary school level. We then discovered as adults that basic artistic skill can be acquired
by just about anyone with functional vision and motor control. Studying classical realism has enabled many of us to “touch the magic” by emphasizing objectivity, accuracy and critical thought. The process of painting and drawing can, at times, seem like little else other than obsessive self-interrogation: “do these two points on the figure really line up vertically?”, “is the figure outline I’ve drawn too thin or too wide?”, “does that shadow shape on my page really look like the one in I see on my subject?”. Why would anyone compromise this aspiration to integrity with mysticism and trivial lies? At its best, art is about truth; and the truth behind the intricacy and complexity of human vision is far more beautiful than any Φ fiction I can come up with. The promotion of mathematical and scientific illiteracy is only useful to those whose aim it is to obscure knowledge.
Social media sites are littered with these gaudy meme-ish images that tell us how we can contract cancer from common foods that are generally considered to be safe: milk, wheat, aspartame, monosodium glutamate or food that isn’t organic. Related images tell us that should we ever find ourselves suffering from this terrible condition, the cures are sitting right under our noses in the forms of green tea, turmeric and “sour sop” fruit. Obviously the reason we don’t find these cancer “cures” being used in hospitals is due to some heartless corporate executive who wants to maintain the profit spread between cancer-causing and cancer-curing foods.
Is there even an iota of truth in any of these memes? The technically correct answer is “at the time of writing, virtually no claim regarding a significant carcinogenic or anti-cancer effect from normal consumption of a generally-regarded-as-safe food is true”. Of course, the technically correct answer doesn’t capture the absurdity behind entire premise here.
Taken together, these memes would have us believe that a cartel of dairy farmers, sugar plantation owners and beverage makers along with “Big Pharma”, the FDA and the entire government take part in a bi-weekly 8am conference call where they discuss clever new ways to carry out their agenda of suppressing the good mangosteen producers of the world. At the end of the call they run back to their offices inside Skull Mountain and The Fortress of Doom, where they gleefully stroke their handlebar mustaches as they direct the distribution of products to promote a single, well-defined disease we call “cancer”. Their ultimate goal, of course, is to enjoy watching people suffer unimaginably in front of their parents, children and siblings.
A mindset that would accept the scenario I have just described as plausible needs to be informed about some other basic truths of the world, like the one about the existence of Santa Claus. These sorts of claims about cancer aren’t merely lies, they are flat out fibs with about as much truth as the proposition that there are monsters under one’s bed. These memes ignore the complex etiology behind a diverse set of diseases and maliciously propagate ignorance of our basic understanding of chemistry, biology and physics. You don’t have to be a professional scientist to ferret out such lies; you just need to be able to tell when someone is trying to con you. Diagnostic markers can often be found in the meme text e.g., prominent quotes with conveniently missing/incomplete references to primary sources (“new study shows that eating dandelions reduces cancer rates by 25%”), a general tone of paranoia (“the secret that cancer doctors don’t want you to know”) as well as very little information with regard to exposure amounts (we are all exposed to carcinogens everyday; the key question when it comes to safety is “how much?”).
While diet certainly contributes to health and cancer susceptibility, there is no evidence that any common and generally available food (“processed”, organic or otherwise) is known to directly cause cancer when consumed in normal, realistic volumes. Similarly, not a single cancer “home-remedy” is part of any therapy that a patient would receive in a hospital today. Even though ongoing studies have taken a close look at a number of home-remedies with some positive preliminary results, no one has found the miraculous curative effects that are often claimed as truth by self-professed practitioners of “alternative medicine”. Government agencies and hospitals who are tasked with improving cancer survival rates and delivering patient care have clearly stated positions that reflect this general sentiment:
So what is the harm if someone wants to drink 10 cups of green tea a day in an attempt to cure or at least prevent a potentially devastating disease? Perhaps there would be none, but when it comes to understanding what really works and what doesn’t, we need to be honest and critical with ourselves so that we can clearly identify underlying causes and therapies that could improve the quality of life and survival rates for patients.
Honest and critical thinking is what has led to the established body of evidence that clearly shows how things like smoking, alcohol (both of which are, sadly, sold in grocery stores), certain viruses and sun exposure are clearly implicated in cancer. Modern medicine is not a panacea when it comes to overall rates of cancer mortality and there is a seldom-discussed cynicism amongst scientists when they reflect on the rate at which new medicines are being discovered (and the funding structure of that entire process). Even so, certain cancer types have enjoyed very significant increases in survival rates (e.g., colon, non-Hodgkins lymphoma and breast) since the mid-1970’s. We would have never realized such improvements if we decided to look at cancer through the lens of pseudoscience.
Boy, it’s hard to write about cancer without thinking about loved ones who I’ve seen suffer dearly at its hand…
Update(06/25/2013): Comments on In The Pipeline inform me that excitotoxicity is a real phenomenon (I was unaware of the precise neurobiological definition), so I should be a little less bugged about cells being “excited to death”; of course, this has nothing to do with dietary intake of glutamate (which the meme implies). Many of these memes use similar tricks, referring to something that is true in a specific technical context but has nothing to do with underlying message they are trying to send.
Update(06/25/2013): Applied a reasonably easy fix to the offending speech bubble.
This is actually a real story that a friend told me about their friend. I wanted to attribute correctly, but after talking to three of my friends, none of them recall telling me. Perhaps it came to me in a dream.
We live in an age of (what I call) “the cult of science”, a generally positive inclination to honest, critical inquiry towards understanding our world. This positivity is not unrelated to some of today’s popular personalities advocating this sort of world view. You’ve probably already heard his radio-quality voice, but have you seen Neil DeGrasse Tyson dance? I think we can safely say we’ve never had scientists this cool.
I think it is important to remember that attitude isn’t sufficient to gain the insights that modern science has acquired; we need the ability allocate massive amounts of capital and future cashflows in concentrated assets representing specialized training and instruments that bear the possibility of little or no return (think of space shuttles, particle accelerators and clinical trials for new drugs).
A lot of times what people mean by “science” isn’t the underlying method, but the government and corporate funding which allows developed countries to perform science at the level they do. A side effect of this system is that research programs are often unveiled with fantastic promises and visions in order to gain the financial resources they require. Sometimes these promises and visions are not realized.
Let me provide you with an example: “nanotechnology”. I cringe when I hear about children wanting to be nanotechnologists when they grow up. Why? After all, there are a number of government programs, scientific journals and foundations which make it seem like nanotech is already here (and we’re just polishing the nano-edges)! The reality is that it is a sort of “victory by declaration”. We originally dreamt of tiny robots doing things like fixing individual cancer cells, but what passes as nanotech today is probably (more fairly) called “chemistry”, “biology” or “material science”. Sure, we are doing some amazing things at the molecular scale these days, but I don’t think we could look back at our dreams and say that we are anywhere close to realizing them yet.
Another aspirationally-named field that comes to mind is “astrobiology”. People really are doing some fantastic work under the funding initiative, but the name makes it sound like we’ve already found life outside our planet (and we haven’t).
The last speech bubble is inspired from the responses of Felisa Wolfe-Simon and Ron Oremland to studies that demonstrated that their super-hyped “arsenic life” paper was plainly wrong. The linked article is a much better recap of the story than what I could provide here, but I found the author’s responses didn’t really acknowledge the fact that carefully observed evidence clearly (and correctly) contradicted their earlier findings which were in error.
Well that was a long break, and both of my readers (well, one if you don’t count Mom) no doubt feel a little ripped off by a stupid Devanagari logo being passed off as a comic (I know, the last letter /ī/ is a bit of a stretch).
Two inspirations for this idea:
1) Nidhi Chanani is doing a weekly “hindi saturday school”. We talked about trying to style Devanagari letters as western music symbols; this is the best I could do (although other letters like “pa”, “va” and “ġa” have pretty obvious interpretations).
2) Something that I haven’t seen nearly enough of: Islamic calligraphy. Although I grew up with (what I thought was) a good understanding of Islamic culture, I only really saw the art form after reading Craig Thompson’s amazingly gorgeous “Habibi”. I also thought I knew something about graphic novels, but I only discovered Craig Thompson after being introduced to him by Nidhi.