When I was in fourth grade, my teacher said to us one day: "There are as many even numbers as there are numbers."
"Really?", I thought.
Well, yeah, there are infinitely many of both, so I suppose there are the same number of them.
But even numbers are only part of the whole numbers, all the odd numbers are left over, so there's got to be more whole numbers than even numbers, right?
To see what my teacher was getting at, let's first think about what it means for two sets to be the same size.
What do I mean when I say I have the same number of fingers on my right hand as I do on left hand?
Of course, I have five fingers on each, but it's actually simpler than that.
I don't have to count, I only need to see that I can match them up, one to one.
In fact, we think that some ancient people who spoke languages that didn't have words for numbers greater than three used this sort of magic.
For instance, if you let your sheep out of a pen to graze, you can keep track of how many went out by setting aside a stone for each one, and putting those stones back one by one when the sheep return, so you know if any are missing without really counting.
As another example of matching being more fundamental than counting, if I'm speaking to a packed auditorium, where every seat is taken and no one is standing, I know that there are the same number of chairs as people in the audience, even though I don't know how many there are of either.
So, what we really mean when we say that two sets are the same size is that the elements in those sets can be matched up one by one in some way.
My fourth grade teacher showed us the whole numbers laid out in a row, and below each we have its double.
As you can see, the bottom row contains all the even numbers, and we have a one-to-one match.
That is, there are as many even numbers as there are numbers.
But what still bothers us is our distress over the fact that even numbers seem to be only part of the whole numbers.
But does this convince you that I don't have the same number of fingers on my right hand as I do on my left?
Of course not.
It doesn't matter if you try to match the elements in some way and it doesn't work, that doesn't convince us of anything.
If you can find one way in which the elements of two sets do match up, then we say those two sets have the same number of elements.
Can you make a list of all the fractions?
This might be hard, there are a lot of fractions!
And it's not obvious what to put first, or how to be sure all of them are on the list.
Nevertheless, there is a very clever way that we can make a list of all the fractions.
This was first done by Georg Cantor, in the late eighteen hundreds.
First, we put all the fractions into a grid.
They're all there.
For instance, you can find, say, 117/243, in the 117th row and 243rd column.
Now we make a list out of this by starting at the upper left and sweeping back and forth diagonally, skipping over any fraction, like 2/2, that represents the same number as one the we've already picked.
We get a list of all the fractions, which means we've created a one-to-one match between the whole numbers and the fractions, despite the fact that we thought maybe there ought to be more fractions.
OK, here's where it gets really interesting.
You may know that not all real numbers -- that is, not all the numbers on a number line -- are fractions.
The square root of two and pi, for instance.
Any number like this is called irrational.
Not because it's crazy, or anything, but because the fractions are ratios of whole numbers, and so are called rationals; meaning the rest are non-rational, that is, irrational.
Irrationals are represented by infinite, non-repeating decimals.
So, can we make a one-to-one match between the whole numbers and the set of all the decimals, both the rationals and the irrationals?
That is, can we make a list of all the decimal numbers?
Cantor showed that you can't.
Not merely that we don't know how, but that it can't be done.
Look, suppose you claim you have made a list of all the decimals.
I'm going to show you that you didn't succeed, by producing a decimal that is not on your list.
I'll construct my decimal one place at a time.
For the first decimal place of my number, I'll look at the first decimal place of your first number.
If it's a one, I'll make mine a two; otherwise I'll make mine a one.
For the second place of my number, I'll look at the second place of your second number.
Again, if yours is a one, I'll make mine a two, and otherwise I'll make mine a one.
See how this is going?
The decimal I've produced can't be on your list.
Why?
Could it be, say, your 143rd number?
No, because the 143rd place of my decimal is different from the 143rd place of your 143rd number.
I made it that way.
Your list is incomplete.
It doesn't contain my decimal number.
And, no matter what list you give me, I can do the same thing, and produce a decimal that's not on that list.
So we're faced with this astounding conclusion: The decimal numbers cannot be put on a list.
They represent a bigger infinity that the infinity of whole numbers.
So, even though we're familiar with only a few irrationals, like square root of two and pi, the infinity of irrationals is actually greater than the infinity of fractions.
Someone once said that the rationals -- the fractions -- are like the stars in the night sky.
The irrationals are like the blackness.
Cantor also showed that, for any infinite set, forming a new set made of all the subsets of the original set represents a bigger infinity than that original set.
This means that, once you have one infinity, you can always make a bigger one by making the set of all subsets of that first set.
And then an even bigger one by making the set of all the subsets of that one.
And so on.
And so, there are an infinite number of infinities of different sizes.
If these ideas make you uncomfortable, you are not alone.
Some of the greatest mathematicians of Cantor's day were very upset with this stuff.
They tried to make these different infinities irrelevant, to make mathematics work without them somehow.
Cantor was even vilified personally, and it got so bad for him that he suffered severe depression, and spent the last half of his life in and out of mental institutions.
But eventually, his ideas won out.
Today, they're considered fundamental and magnificent.
All research mathematicians accept these ideas, every college math major learns them, and I've explained them to you in a few minutes.
Some day, perhaps, they'll be common knowledge.
There's more.
We just pointed out that the set of decimal numbers -- that is, the real numbers -- is a bigger infinity than the set of whole numbers.
Cantor wondered whether there are infinities of different sizes between these two infinities.
He didn't believe there were, but couldn't prove it.
Cantor's conjecture became known as the continuum hypothesis.
In 1900, the great mathematician David Hilbert listed the continuum hypothesis as the most important unsolved problem in mathematics.
The 20th century saw a resolution of this problem, but in a completely unexpected, paradigm-shattering way.
In the 1920s, Kurt Gödel showed that you can never prove that the continuum hypothesis is false.
Then, in the 1960s, Paul J. Cohen showed that you can never prove that the continuum hypothesis is true.
Taken together, these results mean that there are unanswerable questions in mathematics.
A very stunning conclusion.
Mathematics is rightly considered the pinnacle of human reasoning, but we now know that even mathematics has its limitations.
Still, mathematics has some truly amazing things for us to think about.
Meet Lucy.
She was a math major in college, and aced all her courses in probability and statistics.
Which do you think is more likely: that Lucy is a portrait artist, or that Lucy is a portrait artist who also plays poker?
In studies of similar questions, up to 80 percent of participants chose the equivalent of the second statement: that Lucy is a portrait artist who also plays poker.
After all, nothing we know about Lucy suggests an affinity for art, but statistics and probability are useful in poker.
And yet, this is the wrong answer.
Look at the options again.
How do we know the first statement is more likely to be true?
Because it’s a less specific version of the second statement.
Saying that Lucy is a portrait artist doesn’t make any claims about what else she might or might not do.
But even though it’s far easier to imagine her playing poker than making art based on the background information, the second statement is only true if she does both of these things.
However counterintuitive it seems to imagine Lucy as an artist, the second scenario adds another condition on top of that, making it less likely.
For any possible set of events, the likelihood of A occurring will always be greater than the likelihood of A and B both occurring.
If we took a random sample of a million people who majored in math, the subset who are portrait artists might be relatively small.
But it will necessarily be bigger than the subset who are portrait artists and play poker.
Anyone who belongs to the second group will also belong to the first– but not vice versa.
The more conditions there are, the less likely an event becomes.
So why do statements with more conditions sometimes seem more believable?
This is a phenomenon known as the conjunction fallacy.
When we’re asked to make quick decisions, we tend to look for shortcuts.
In this case, we look for what seems plausible rather than what is statistically most probable.
On its own, Lucy being an artist doesn’t match the expectations formed by the preceding information.
The additional detail about her playing poker gives us a narrative that resonates with our intuitions— it makes it seem more plausible.
And we choose the option that seems more representative of the overall picture, regardless of its actual probability.
This effect has been observed across multiple studies, including ones with participants who understood statistics well– from students betting on sequences of dice rolls, to foreign policy experts predicting the likelihood of a diplomatic crisis.
The conjunction fallacy isn’t just a problem in hypothetical situations.
Conspiracy theories and false news stories often rely on a version of the conjunction fallacy to seem credible– the more resonant details are added to an outlandish story, the more plausible it begins to seem.
But ultimately, the likelihood a story is true can never be greater than the probability that its least likely component is true.
There are two groups of women when it comes to screening mammography -- women in whom mammography works very well and has saved thousands of lives and women in whom it doesn't work shale@example.net.
Do you know which group you're in?
If you don't, you're not alone.
Because the breast has become a very political organ.
The truth has become lost in all the rhetoric coming from the press, politicians, radiologists and medical imaging companies.
I will do my best this morning to tell you what I think is the truth.
But first, my disclosures.
I am not a breast cancer survivor.
I'm not a radiologist.
I don't have any patents, and I've never received any money from a medical imaging company, and I am not seeking your vote.
(Laughter) What I am is a doctor of internal medicine who became passionately interested in this topic about 10 years ago when a patient asked me a question.
She came to see me after discovering a breast lump.
Her sister had been diagnosed with breast cancer in her 40s.
She and I were both very pregnant at that time, and my heart just ached for her, imagining how afraid she must be.
Fortunately, her lump proved to be benign.
But she asked me a question: how confident was I that I would find a tumor early on her mammogram if she developed one?
So I studied her mammogram, and I reviewed the radiology literature, and I was shocked to discover that, in her case, our chances of finding a tumor early on the mammogram were less than the toss of a coin.
You may recall a year ago when a firestorm erupted after the United States Preventive Services Task Force reviewed the world's mammography screening literature and issued a guideline recommending against screening mammograms in women in their 40s.
Now everybody rushed to criticize the Task Force, even though most of them weren't in anyway familiar with the mammography studies.
It took the Senate just 17 days to ban the use of the guidelines in determining insurance coverage.
Radiologists were outraged by the guidelines.
The pre-eminent mammographer in the United States issued the following quote to the Washington Post.
The radiologists were, in turn, criticized for protecting their own financial self-interest.
But in my view, the radiologists are heroes.
There's a shortage of radiologists qualified to read mammograms, and that's because mammograms are one of the most complex of all radiology studies to interpret, and because radiologists are sued more often over missed breast cancer than any other cause.
But that very fact is telling.
Where there is this much legal smoke, there is likely to be some fire.
The factor most responsible for that fire is breast density.
Breast density refers to the relative amount of fat -- pictured here in yellow -- versus connective and epithelial tissues -- pictured in pink.
And that proportion is primarily genetically determined.
Two-thirds of women in their 40s have dense breast tissue, which is why mammography doesn't work as well in them.
And although breast density generally declines with age, up to a third of women retain dense breast tissue for years after menopause.
So how do you know if your breasts are dense?
Well, you need to read the details of your mammography report.
Radiologists classify breast density into four categories based on the appearance of the tissue on a mammogram.
If the breast is less than 25 percent dense, that's called fatty-replaced.
The next category is scattered fibroglandular densities, followed by heterogeneously dense and extremely dense.
And breasts that fall into these two categories are considered dense.
The problem with breast density is that it's truly the wolf in sheep's clothing.
Both tumors and dense breast tissue appear white on a mammogram, and the X-ray often can't distinguish between the two.
So it's easy to see this tumor in the upper part of this fatty breast.
But imagine how difficult it would be to find that tumor in this dense breast.
That's why mammograms find over 80 percent of tumors in fatty breasts, but as few as 40 percent in extremely dense breasts.
Now it's bad enough that breast density makes it hard to find a cancer, but it turns out that it's also a powerful predictor of your risk for breast cancer.
It's a stronger risk factor than having a mother or a sister with breast cancer.
At the time my patient posed this question to me, breast density was an obscure topic in the radiology literature, and very few women having mammograms, or the physicians ordering them, knew about this.
But what else could I offer her?
Mammograms have been around since the 1960's, and it's changed very little.
There have been surprisingly few innovations, until digital mammography was approved in 2000.
Digital mammography is still an X-ray of the breast, but the images can be stored and manipulated digitally, just like we can with a digital camera.
The U.S. has invested four billion dollars converting to digital mammography equipment, and what have we gained from that investment?
In a study funded by over 25 million taxpayer dollars, digital mammography was found to be no better over all than traditional mammography, and in fact, it was worse in older women.
But it was better in one group, and that was women under 50 who were pre-menopausal and had dense breasts, and in those women, digital mammography found twice as many cancers, but it still only found 60 percent.
So digital mammography has been a giant leap forward for manufacturers of digital mammography equipment, but it's been a very small step forward for womankind.
What about ultrasound?
Ultrasound generates more biopsies that are unnecessary relative to other technologies, so it's not widely used.
And MRI is exquisitely sensitive for finding tumors, but it's also very expensive.
If we think about disruptive technology, we see an almost ubiquitous pattern of the technology getting smaller and less expensive.
Think about iPods compared to stereos.
But it's the exact opposite in health care.
The machines get ever bigger and ever more expensive.
Screening the average young woman with an MRI is kind of like driving to the grocery store in a Hummer.
It's just way too much equipment.
One MRI scan costs 10 times what a digital mammogram costs.
And sooner or later, we're going to have to accept the fact that health care innovation can't always come at a much higher price.
Malcolm Gladwell wrote an article in the New Yorker on innovation, and he made the case that scientific discoveries are rarely the product of one individual's genius.
Rather, big ideas can be orchestrated, if you can simply gather people with different perspectives in a room and get them to talk about things that they don't ordinarily talk about.
It's like the essence of TED.
He quotes one innovator who says, "The only time a physician and a physicist get together is when the physicist gets sick."
(Laughter) This makes no sense, because physicians have all kinds of problems that they don't realize have solutions.
And physicists have all kinds of solutions for things that they don't realize are problems.
Now, take a look at this cartoon that accompanied Gladwell's article, and tell me if you see something disturbing about this depiction of innovative thinkers.
(Laughter) So if you will allow me a little creative license, I will tell you the story of the serendipitous collision of my patient's problem with a physicist's solution.
Shortly after her visit, I was introduced to a nuclear physicist at Mayo named Michael O'Conner, who was a specialist in cardiac imaging, something I had nothing to do with.
And he happened to tell me about a conference he'd just returned from in Israel, where they were talking about a new type of gamma detector.
Now gamma imaging has been around for a long time to image the heart, and it had even been tried to image the breast.
But the problem was that the gamma detectors were these huge, bulky tubes, and they were filled with these scintillating crystals, and you just couldn't get them close enough around the breast to find small tumors.
But the potential advantage was that gamma rays, unlike X-rays, are not influenced by breast density.
But this technology could not find tumors when they're small, and finding a small tumor is critical for survival.
If you can find a tumor when it's less than a centimeter, survival exceeds 90 percent, but drops off rapidly as tumor size increases.
But Michael told me about a new type of gamma detector that he'd seen, and this is it.
It's made not of a bulky tube, but of a thin layer of a semiconductor material that serves as the gamma detector.
And I started talking to him about this problem with breast density, and we realized that we might be able to get this detector close enough around the breast to actually find small tumors.
So after putting together a grid of these cubes with tape -- (Laughter) -- Michael hacked off the X-ray plate of a mammography machine that was about to be thrown out, and we attached the new detector, and we decided to call this machine Molecular Breast Imaging, or MBI.
This is an image from our first patient.
And you can see, using the old gamma technology, that it just looked like noise.
But using our new detector, we could begin to see the outline of a tumor.
So here we were, a nuclear physicist, an internist, soon joined by Carrie Hruska, a biomedical engineer, and two radiologists, and we were trying to take on the entrenched world of mammography with a machine that was held together by duct tape.
To say that we faced high doses of skepticism in those early years is just a huge understatement, but we were so convinced that we might be able to make this work that we chipped away with incremental modifications to this system.
This is our current detector.
And you can see that it looks a lot different.
The duct tape is gone, and we added a second detector on top of the breast, which has further improved our tumor detection.
So how does this work?
The patient receives an injection of a radio tracer that's taken up by rapidly proliferating tumor cells, but not by normal cells, and this is the key difference from mammography.
Mammography relies on differences in the appearance of the tumor from the background tissue, and we've seen that those differences can be obscured in a dense breast.
But MBI exploits the different molecular behavior of tumors, and therefore, it's impervious to breast density.
After the injection, the patient's breast is placed between the detectors.
And if you've ever had a mammogram -- if you're old enough to have had a mammogram -- you know what comes next: pain.
You may be surprised to know that mammography is the only radiologic study that's regulated by federal law, and the law requires that the equivalent of a 40-pound car battery come down on your breast during this study.
But with MBI, we use just light, pain-free compression.
(Applause) And the detector then transmits the image to the computer.
So here's an example.
You can see, on the right, a mammogram showing a faint tumor, the edges of which are blurred by the dense tissue.
But the MBI image shows that tumor much more clearly, as well as a second tumor, which profoundly influence that patient's surgical options.
In this example, although the mammogram found one tumor, we were able to demonstrate three discrete tumors -- one is small as three millimeters.
Our big break came in 2004.
After we had demonstrated that we could find small tumors, we used these images to submit a grant to the Susan G. Komen Foundation.
And we were elated when they took a chance on a team of completely unknown investigators and funded us to study 1,000 women with dense breasts, comparing a screening mammogram to an MBI.
Of the tumors that we found, mammography found only 25 percent of those tumors.
MBI found 83 percent.
Here's an example from that screening study.
The digital mammogram was read as normal and shows lots of dense tissue, but the MBI shows an area of intense uptake, which correlated with a two-centimeter tumor.
In this case, a one-centimeter tumor.
And in this case, a 45-year-old medical secretary at Mayo, who had lost her mother to breast cancer when she was very young, wanted to enroll in our study.
And her mammogram showed an area of very dense tissue, but her MBI showed an area of worrisome uptake, which we can also see on a color image.
And this corresponded to a tumor the size of a golf ball.
But fortunately it was removed before it had spread to her lymph nodes.
So now that we knew that this technology could find three times more tumors in a dense breast, we had to solve one very important problem.
We had to figure out how to lower the radiation dose, and we have spent the last three years making modifications to every aspect of the imaging system to allow this.
And I'm very happy to report that we're now using a dose of radiation that is equivalent to the effective dose from one digital mammogram.
And at this low dose, we're continuing this screening study, and this image from three weeks ago in a 67-year-old woman shows a normal digital mammogram, but an MBI image showing an uptake that proved to be a large cancer.
So this is not just young women that it's benefiting.
It's also older women with dense tissue.
And we're now routinely using one-fifth the radiation dose that's used in any other type of gamma technology.
MBI generates four images per breast.
MRI generates over a thousand.
It takes a radiologist years of specialty training to become expert in differentiating the normal anatomic detail from the worrisome finding.
But I suspect even the non-radiologists in the room can find the tumor on the MBI image.
But this is why MBI is so potentially disruptive -- it's as accurate as MRI, it's far less complex to interpret, and it's a fraction of the cost.
But you can understand why there may be forces in the breast-imaging world who prefer the status quo.
After achieving what we felt were remarkable results, our manuscript was rejected by four journals.
After the fourth rejection, we requested reconsideration of the manuscript, because we strongly suspected that one of the reviewers who had rejected it had a financial conflict of interest in a competing technology.
Our manuscript was then accepted and will be published later this month in the journal Radiology.
(Applause) We still need to complete the screening study using the low dose, and then our findings will need to be replicated at other institutions, and this could take five or more years.
If this technology is widely adopted, I will not benefit financially in any way, and that is very important to me, because it allows me to continue to tell you the truth.
But I recognize -- (Applause) I recognize that the adoption of this technology will depend as much on economic and political forces as it will on the soundness of the science.
The MBI unit has now been FDA approved, but it's not yet widely available.
So until something is available for women with dense breasts, there are things that you should know to protect yourself.
First, know your density.
Ninety percent of women don't, and 95 percent of women don't know that it increases your breast cancer risk.
The State of Connecticut became the first and only state to mandate that women receive notification of their breast density after a mammogram.
I was at a conference of 60,000 people in breast-imaging last week in Chicago, and I was stunned that there was a heated debate as to whether we should be telling women what their breast density is.
Of course we should.
And if you don't know, please ask your doctor or read the details of your mammography report.
Second, if you're pre-menopausal, try to schedule your mammogram in the first two weeks of your menstrual cycle, when breast density is relatively lower.
Third, if you notice a persistent change in your breast, insist on additional imaging.
And fourth and most important, the mammography debate will rage on, but I do believe that all women 40 and older should have an annual mammogram.
Mammography isn't perfect, but it's the only test that's been proven to reduce mortality from breast cancer.
But this mortality banner is the very sword which mammography's most ardent advocates use to deter innovation.
Some women who develop breast cancer die from it many years later, and most women, thankfully, survive.
So it takes 10 or more years for any screening method to demonstrate a reduction in mortality from breast cancer.
Mammography's the only one that's been around long enough to have a chance of making that claim.
It is time for us to accept both the extraordinary successes of mammography and the limitations.
We need to individualize screening based on density.
For women without dense breasts, mammography is the best choice.
But for women with dense breasts; we shouldn't abandon screening altogether, we need to offer them something better.
The babies that we were carrying when my patient first asked me this question are now both in middle school, and the answer has been so slow to come.
She's given me her blessing to share this story with you.
After undergoing biopsies that further increased her risk for cancer and losing her sister to cancer, she made the difficult decision to have a prophylactic mastectomy.
We can and must do better, not just in time for her granddaughters and my daughters, but in time for you.
Thank you.
(Applause)