Meditations on “infinity” (with apologies for taking liberties in the Holiday spirit)

Les deux infinis 

Infinity has at least two meanings. We sometimes forget that the word is used very differently in math and in science, and we conflate the two.

For the mathematician, infinity is an abstraction with a pure and definite meaning, with some but not all the properties of a number. There is the magic that says that the infinity of even integers is exactly the same as the infinity of all integers, even and odd. More amazing is that the count of all fractional numbers like 1.65 and 3¾ and 0.142857142857142857… is also this same infinity. Yet there are infinitely many different infinities, labeled א_o, א_l, etc. and in the same sense, each can be demonstrated to be not equal, but “larger” than the one that precedes it.

But in the world of science, this kind of infinity doesn’t concern us. No experiment that can be performed will ever yield the result, “infinity”. There is only “beyond our ability to measure at this time”, and the related concept, “large enough that we need not worry about its effect on such-and-such a calculation.” Infinity has a practical meaning, relative to any given measurement or calculation.

We are finite creatures. All our experimental apparati are finite. Empirically, there is no such thing as “infinity”, but only a human-scaled limit for our ability to measure or calculate.

Does space go on infinitely in all directions, or is it merely larger than anything that we can measure (or that we can calculate from the measurements we make)? Mathematically, this may be a question we wish to ask, but scientifically it is not a meaningful question because it is not a question to which we can assign any empirical meaning.

Similarly, there is a mathematical meaning, but no physical meaning to the concept eternity.

We may wish for eternal life, but we cannot conceive eternity. Homo sapiens has been on earth for a mere 200,000 years, and even that is so far beyond our experience as to be an abstraction to us.

We like to think of “getting death out of the picture”, so we need never worry about dying. Think of it as a lifetime so long that, in practice, you just don’t worry about its limits. But don’t think about eternity – the very idea is full of paradoxes.

Start with the fact that you have a finite brain with finite memory capacity. Eventually, it would all get used up. You wouldn’t have the capacity to remember most of the stuff that has happened to you. (The mathematician would say, “as the duration of life increases without bound, the fraction of total life experience that remains stored in memory becomes vanishingly small.”)

One of the weirdnesses of quantum mechanics is that physical states are not continuous, but discrete and countable. The number of physical ways in which any finite amount of matter can be arranged is an unimaginably large number, but not mathematically infinite. If you lived forever, you would eventually go through every possible experience, and then you would have every possible experience again, and again and again…and you would have, at that point, barely begun.

So if we lived forever, we’d run out of experiences. For a finite being, there can be no eternal life, but only infinitely many repetitions from a finite set of experiences. I’m reminded of Pär Lagerqvist’s character who, for a momentary indiscretion, draws a curse from the condemned man bearing his cross: “You shall never die!”

Perhaps our wish for eternal life is no more than a paraphrase of our fear of death. Perhaps this is better addressed with attention to dissolving fear than to banishing death.

Of aging and eternal life

The fountain of youth has been a staple of stories, dreams and frauds for the duration of human history, but in the latter half of the 20^th century the human life span has actually been extended by technology for the first time. Just in our generation, it has become possible to contemplate a thorough prevention of aging through medical intervention.

The absence of aging is not the same as immortality. At its low point, actuarial risk of death (for 10-year-old girls) is about 1 in 1,000 per year. So if we were never to age at all, perhaps we would live an average of 1,000 years.

This would be an inconceivably wonderful thing, at least for us as individuals. The danger of disease would be slashed, and the weakening and loss of faculties that come with age would be eliminated as well. For our human community, the effects would be more complex, probably with mixed benefits and risks. Imagine the wealth of human experience that would be available in the memories of our elders. Imagine the adaptability to change (phenotypic plasticity is the biological word) that would be demanded of us over that long a span of time, assuming we choose to remain engaged, rather than, say, to work for 60 years then retire for 940. Think how difficult it is for people educated and trained before computers to adapt to today’s work environments.  Consider that this represents only 30 years, and that the pace of change is quickening, some say at an exponential rate.

 

1,000 years? I could get used to that

I find it utterly bewildering to contemplate eternity. I lose all perspective, and context for what I am doing in the present moment which (as Buddhists are fond of saying) is all we have.

In contrast, when I think of a lifetime of 1,000 years, I breathe a sigh, my imagination opens in scope, I experience a broadening perspective, empowerment, a loosening of anxiety about the immediate future and a freedom to experiment.

The short-term arc of the news cycle can feel sickening, alienating, arbitrary, even surreal in its lack of meaning or direction. But the long arc of history bends toward justice.  Imagine the freedom to participate in that long arc, to be part of a larger human endeavor that spans lifetimes.

Imagine opening the paper and reading commentary on current events from Thomas Jefferson or Oliver Wendell Holmes.

If you had the opportunity to meet Leonardo, what might you like to ask him?

Imagine the privilege of presenting Beethoven with an iPod containing his 9 symphonies, 32 sonatas and 16 string quartets (though perhaps we would be wise to offer him first a cochlear implant).

Imagine explaining the power of nonviolence or the importance of scientific integrity to your 7th-great grandchild.

The biggest hazard of the 1,000-year life span is the spectre of tyrants clinging to power. Political institutions need to be able to adapt, and scientific understandings even more so. If ‘science progresses funeral by funeral’ because well-established scientists become gatekeepers with a stake in old ideas, then how will scientific institutions have to adapt in order to promote progress when people are living through multiple scientific revolutions?

These were idle fantasies in the past, but they have become dreams sanctioned by science.

Embracing mystery

Without aging, our lives will not only be much longer, but also of less definite duration. When life spans are fixed by aging, to live 20% beyond the average is an upper limit. No one yet has lived twice the average life span. But without aging, all deaths would be “accidental” deaths, and 1,000 years would be only the mean of a broad, exponential distribution. One person in 7 would live more than 2,000 years and one in 20 would live more than 3,000 years. With billions of people on earth, there would be many who had lived 10,000 years, and someone, somewhere who could remember 15,000 years ago.

To live without fixed duration is a good thing, precisely because it frustrates our propensity to make fixed plans. The blueprints for our lives, with limited career goals, limited horizons and planned retirement dates, are constraining in global ways which we do not apprehend because we are fish swimming in water.

We realize easily that it would be a great burden to know in advance the day on which we will die; the more approximate sense in which we know our life expectancies is also hurtful to us. Life should be open-ended. We have all heard stories, in one form or another, of the person who has planned for a fixed length to his years, then reaches the end of his plan and finds himself with blood still in his veins. At first relieved, then increasingly confused, he asks, “why am I still alive?”

Opening into unknown destinies is the essence of our being. Though our bodies have a finite duration, our creative spirits can thrive best at the threshold of the infinite.

…not in the mathematical sense, of course.

 For basic information about healthy living for a long life,
see the author’s permanent page at AgingAdvice.org.

In the news this week was a huge British study in The Lancet about longevity trends worldwide. The headlines were about the admirable progress that our world is making toward fewer avoidable deaths in the underdeveloped world. Life expectancy there is going up, for reasons that have nothing to do with aging. The secondary headline: Life expectancy in the US is falling further behind other rich countries, and this is much more true of American women than American men. This is a national humiliation, and it has everything to do with economics.

 

Good news for the underdeveloped world

From 1970 to 2010, life expectancy the world over increased from 59 to 70. Threescore and ten is no longer the province of the rich, but an expectation for our great human family. Deaths of children younger than 5 years declined in absolute terms by almost 60% since 1970, despite a doubling in world population.

 

In the developed countries, it is about aging.

In 1970, in the developed world, the low-hanging fruit had already been picked clean: infectious disease, childbirth, and infant mortality were no longer major factors in actuarial risk. Most people could expect to live out their full life span of about 70 years. Everyone expected – that is, demographers, epidemiologists and policy-makers – everyone expected that the life expectancy had risen to a natural limit, and that any further progress would be slow and difficult. But surprise! Since 1970, countries with the longest life expectancies have continued to improve just as fast as before, even though the progress is now all “at the back end”. Increase in life expectancy since 1970 has been more than 10 years in Japan, 9 years in Europe. For every year that goes by, 3 months are added to life expectancy. And this progress applies almost entirely to people over 70. Seniors today are living longer because they are healthier. They are more engaged and active than their parents’ generation.

Compare a picture of a 60-year-old in 1912 to a 70-year-old today. James Vaupel is the world’s most famous, if controversial, demographer. (He is not among the authors of the Lancet study.)  By his definition, we have succeeded in delaying the aging process by a decade.

The evidence suggests that deterioration, instead of being stretched out, is being postponed: levels of mortality and other indices of health that used to prevail at age 70 now prevail at age 80, and levels that used to prevail at age 80 now prevail at age 90.

Exactly “how” is a mystery, but there is wide agreement that it has been a composite of many factors. “It seems that death is being delayed because people are reaching old age in better health.”

 

Bad news for US Women

In 1970, both Europe and Japan had recently slipped by the US as leaders in life expectancy. The trend has continued for 40 years, and women in Japan now live 5½ years longer than American women. For men the gap is 3 years. European women and men are halfway between their counterparts in the US and Japan.

(These numbers come from the life expectancy table by country that is the primary result of the Lancet study.)

Life expectancy in the US is increasing much more slowly than in other parts of the world, and for women, it’s hardly improving at all. The New York Times reports

 But while developing countries made big strides the United States stagnated. American women registered the smallest gains in life expectancy of all high-income countries’ female populations between 1990 and 2010. American women gained just under two years of life, compared with women in Cyprus, who lived 2.3 years longer and Canadian women who gained 2.4 years. The slow increase caused American women to fall to 36th place in the report’s global ranking of life expectancy, down from 22nd in 1990. Life expectancy for American women was 80.5 in 2010, up from 78.6 in 1990.

A lot of the reason for the difference comes from economics. There is a much wider gap in the US than anywhere else in the developed world in access to health care, and this shows up as a major gap in life expectancy between rich and poor. Life expectancy for the working class in America lags the rich by 5 years.

But lest rich Americans should feel complacent about this situation, the Japanese have pulled ahead of the US by more than 6 years. That means that the average Japanese is living longer than the richest Americans.

 

Footnote: What do life expectancy tables mean?

This is a much more interesting and complicated question than it ought to be. The issue is that the world is changing so fast, and any “snapshot” of mortality rates at different ages reflects people who were born at different times. Life expectancy numbers as quoted are not the life expectancy of an infant born today, or for any particular person or set of people. That’s because the numbers are calculated using a composite of all the people dying today. So the standard tables are based on infant mortality of babies just born, combined with war statistics, suicides and traffic accidents for young men born in the 1980s and geriatric statistics for people born in the 1920s.

Based on Vaupel’s claim that 1 year of life is being added for every four years that passes, we might make a very crude correction to tables of life expectancy by tacking on an extra 1/3. If the life table says you have 30 years left to live, you can interpret that to mean 40. (Why is it 1/3 rather than ¼? The answer is purely mathematical, and derives from the fact that while you’re living out your ¼, the world is advancing and giving you another ¼ of ¼, and then while you’re living out that period…etc.)

So what is the life expectancy of a child born today? That’s a question that is as difficult to forecast as anything else about the coming century – war and peace, global warming, the Apocalypse or the Singularity. Based purely on demographic trends and not biomedical forecasts, Vaupel says “The future is uncertain, but it seems plausible that very long lives may be the probable destiny of younger people alive today.”

The best theory we have for the evolution of aging says that bearing children should shorten your life span. The best theory is wrong.

Current Evolutionary Theory

Here’s the surprise from genetic research in the 1990s that changed the way evolutionary scientists think about aging: Aging takes place under control of an intricate regulatory system, ultimately governed by genes. What is more, some of these genes have been conserved over a huge range of life forms, dating back to the dawn of multi-cellular life. No one had anticipated this, but once it was established, there could be no more talk of aging as a passive process, the body “wearing out” like an old shoe or a rusty car.

Then the question arose, what keeps these genes in the genome? Why have they not been eliminated by natural selection long ago? Darwinian evolution is supposed to select only what’s good for you, and aging is bad for you.

So evolutionary scientists hypothesized that aging is caused by “tradeoffs”. In order to maximize fitness, natural selection has made a sort of devil’s bargain, accepting deterioration and death in return for enhanced fitness. In the most prominent theory of our day, enhanced fitness comes in the form of fertility.

This theory – called “Antagonistic Pleiotropy” – predicts that fertility and longevity are genetically and metabolically linked on a see-saw, so that more of one means less of the other. The most direct and important consequence of the theory is that having children ought to accelerate aging.

 

Testing predictions of the theory

Scientists have looked and looked for this effect, to no avail. They’ve studied zoo animals and lab animals, modern medical records, and historical records going back hundreds of years. All these studies have come up negative. Everywhere they look, zoologists and demographers have all found there is a small positive correlation between fertility and longevity.

There is one exception. For a woman, giving birth after the age of 40 shows a large positive correlation. Bearing children after age 40 adds 3 years to a woman’s life expectancy.

From the lab:

In the UC Irvine laboratory of Michael Rose, fruitflies were artificially selected for increased longevity. Year after year, their life span increased from 2 week to 3, then 4 and beyond. Theory said that as life span increased, fertility would have to go south. But in fact, Rose found that fertility was actually rising with longevity. (He has explained the result as an artifact of inadvertent selection for fertility. I say that no matter how the experiment was conducted, you’re not supposed to be able to simultaneously increase fertility and longevity. His results contradict the Antagonistic Pleiotropy theory.)

In studies conducted at UCSF and University College  London and McGill, worms have been genetically modified to live longer, sometimes with no effect on their fertility.

 

From animals in captivity:

Unlike animals in the wild, zoo animals are protected from disease, famine and starvation, so they typically live long enough to die of old age. Robert Ricklefs solicited zoo records from around the world and analyzed 18 species of mammals and 12 birds. He sliced and diced the data 68 different ways, and the only significant correlations he found were positive. He concludes that there is no accelerated aging from laying eggs or bearing young.

 

From historic studies of humans:

In a historic study of French Canadian women in the 17^th and 18^th centuries, a small positive association was found between the number of children they had and their age at death. A historic study of Amish women found the same thing. A 200-year Finnish study found a positive correlation and here is another historic study from a German database.

Natalia Gavrilova and her husband Leonid Gavrilov have been collecting and analyzing data on aging at University of Chicago for many years. Their book chapter summarizes these and other historic studies. To be fair, a lot of women died in childbirth before the 20^th century, and all these studies have had to separate the immediate risk of fatal complications from any long-term effect on the rate of aging.

The remarkable thing I find in reviewing these papers is how consistent they are, and yet each one reports the results as if it were a unique surprise. It seems that no matter how often countervailing data are confirmed, expectations are still controlled by the theory.

 

Contemporary studies of humans:

This recent study of Norwegian women found a positive relationship between fertility and longevity, and reported it as though it were a departure from expectation.

Just last month, a new study of 16,000 Swedish twins reports that childless men and women lived longer than their twins who had children. The three authors from Univ of Southern California mince no words in announcing that their results contradict the pleiotropy theories.

And most amusing is the “dirty old man” study, published with a stiff upper lip in the British Medical Journal a few years ago. Sexual activity for men was found to be positively associated with longevity.

 

The way science works, and the way science is supposed to work

The British scientist who originated one of the tradeoff theories himself did a historic study of British aristocracy, with records going back to the 11^th century. This was the only study to report a confirmation of the theory. For a small subset of the women, using an inappropriate statistical test, he was able to force the data to corroborate his expectation. The study has been severely criticized by others besides me. But the funny thing is that this positive study was reported prominently in Nature, and it has been cited more times than all the negative studies I’ve mentioned combined.

In a healthy scientific community, theories that make the wrong prediction time after time are discredited. But this can take a generation or more. In the meanwhile, the “best theory we have” about the evolution of aging is not much use to us. Still, literature in the field continues to interpret new results in the context of Antagonistic Pleiotropy. It was Max Planck who warned us, “science progresses funeral by funeral.”

One result that stands out

Thomas Perls of Harvard Medical School recruits centenarians and has compiled a database of their genetics, their relatives, and their life styles. The earliest and most striking thing he found was that women who live to be 100 are 4 times more likely to have given birth to a child after age 40 than other women born in that time period. Because of the theory, he was reluctant to argue for a causal effect. He tried to explain the correlation qualitatively by saying that women who are still fertile at 40 are more likely to have the longevity genes that enable them to live to 100. I have argued that this explanation doesn’t work quantitatively, because there are too many women who are fertile at 40 and too few centenarians. My interpretation also agrees with the French Canadian study mentioned above. I believe that having a child when you are past 40 actually increases your life span, and I calculate that the benefit is more than 3 years.

There’s a social stigma attached to being fat, a cultural prejudice that is utterly unfair. It makes no distinction between those who are lax about their diet and those whose metabolisms incline them toward obesity even when they eat moderately. But Mother Nature is more equitable. She distributes the gift of longevity not according to weight, but according to how much you eat, taking body type into account.  And herein lies a resolution to a long-standing paradox of epidemiology.

 

The Paradox

In experiments with animals, there is a very near linear relationship between food intake and life span. The less they eat, the longer they live. Here, for example, are some typical results with mice:

Notice that at the left end, the mice are eating less than a third of what they would like to eat if left to their own devices, but there is no sign of the curve flattening. In fact, experiments have shown the rate of aging continues to be slowed by caloric restriction right up to the threshold of starvation.

But in humans, there seems to be an “ideal weight” where the risk of mortality is minimum. BMI (for Body Mass Index) is a standard measure of weight relative to height. It’s certainly true that people with high BMI are at increased risk of cancer and heart disease, and their life expectancies are shortened. You might expect that people with the lowest BMI had the lowest mortality. But this is not so. Below BMI of 25, there appears to be no benefit from lower weight. For BMI below about 21, the mortality actually starts to climb, and is distinctly higher for BMI less than 18.

BMI of 25 is not especially thin. It corresponds to a man (say 5 foot 10) who weighs 174, or a woman (say 5 foot 4) who weighs 145. (Values are similar in these two recent studies (2010) and (2009)).

There are some studies that suggest even higher values. For women over 70, there’s a study that found the minimum mortality at BMI=32, and for men over 70 it was BMI=29. (1997) Corresponding weights are 223 for men and 179 for women.

What’s going on? Are people so different from other animals, then?

 

Habits and Heredity

Humans are genetically diverse. For any given person, his or her weight is determined by a combination of genetics and life style, including diet and exercise habits. A person may be slender because she has iron discipline, works out regularly and keeps to a low calorie diet; or she may one of those people who can eat whatever she wants and never get fat. Looking at the two people side-by-side, you might not be able to tell which was which; but their metabolic stats would probably tell their tales. Insulin sensitivity is likely to be better in the second person, and markers of inflammation probably higher in the first.

Here’s a clue from a 1983 study at Jackson Labs in Maine. (For those of you who are not familiar with them, Jackson Lab is the world’s leading supplier of lab mice, in many different strains, specialized by breeding or genetic modifications. Profits from selling lab mice are used to support a research agenda that would be the envy of any university.)

David Harrison compared normal mice to mice that carried two copies of a gene for obesity (ob/ob). He put groups of mice of each genetic type on various caloric regimes. The ob/ob mice that were on restricted diets still looked fat, and in fact they had 50% body fat. Normal mice on a similar diet had only 13% body fat. But the two groups showed comparable life extension – both about 50% ahead of normal. In fact, the ob/ob mice were less vulnerable to starvation, so they could survive with less than the normal mice. The group that lived longest was ob/ob mice on severe calorie restriction, but they looked like normal mice that were fully fed.

(Active and productive in his 70’s, Harrison still leads a research group at Jackson Labs. Can you guess his body type?)

This suggests that fat can have a very different metabolic significance, depending on genetics and body type.

 

My theory

Here’s what I think is going on in people. The reason that you don’t see consistently longer life expectancies with lower and lower BMI is that BMI is mostly determined by genes, not life style.  Some people look to be quite overweight, but are healthy nonetheless. People with low values of BMI are probably that way because of their genes, not their habits. “Skinny genes” don’t improve your health or longevity statistics. Only eating less can do that.

What is more, our prejudice against fat tends to magnify this apparent paradox. In our culture, people (especially women) are highly motivated to control their diets when their genes tend to make them overweight. Those whose genes are “more forgiving”, are likely to be lax in their habits – they figure that weight isn’t a problem form them, so they’re less likely to cut calories or to work out.

A lot of those people with BMI of 25 or more are genetically programmed to be heavier than they are, but they’re working hard to keep their weight down. The reason that the health benefit of being skinny doesn’t show up in the statistics is that the higher up the BMI scale you look, the more likely you are to find people who watch their calories.

The reason that the “optimal” BMI for people over 70 appears to be so high is that people who weigh 223 pounds don’t make it to age 70 unless they have good genes. The ones who survive to qualify for the study are a filtered sample.

The reason that the “optimal” BMI is higher for women than men is that social pressure to diet is greater for women.

 

Bottom line for health

The moral of this story is that we shouldn’t be lulled into laxity by the fact that there doesn’t appear to be a longevity benefit for lowering BMI past 25. For any given individual, it’s probably true that the less you eat the longer you live. But this fact is masked in the demographic statistics for two reasons. First that genetics is more important than life style in determining BMI; and second, there is an inverse correlation between food intake and genetic propensity to be stout (as encouraged by our culture).

Everyone can benefit from keeping weight down. But the difference seems to be greater for people who are genetically stout. People who are cursed with an “efficient” metabolism that stores fat get the most benefit from diet and exercise.