This is also a new age of openness and deception. The increase in both is of course coupled. As openness is supported or exhibited, some of what is exposed resists and retaliates with deeper forms of deception.

And in ths new age, the old illusions also persist — like the illusions of rationality or unbiased examination or study without preconceived ideas — illusions that have a great impact on inference and our abilities to draw conclusions from observations. Democracy enters when we attempt to create a cooperative or civil society based in some semblance of truth or grasp of reality.

**The problems I am focused on in this perhaps too provocatively titled article**, are those caused by the use of data and democracy as tools of forceful persuasion or even hammers of coercion. While the idea that democracy is a system predisposed to tyranny is far from a new idea, the dangers in the new bandwagon of data-driven thinking seem to be less well known or thought about (even though Cathy O’Neil’s, *Weapons of Math Destruction* is a good start). So we will begin there.

**It might be seem strange for someone who is a mathematician**, with a great deal of experience in data science, who even founded the *Data Driven Modeling and Analysis* team at Los Alamos National Laboratory, to be concerned with or ambivalent about data driven anything.

Yet I am.

In fact I am very concerned. And the source of the concern is the inescapable fact that every inference, every conclusion and policy that is derived from data, is extracted from the data through the use of prior assumptions, many of which are unacknowledged or even very difficult to see. We can begin with the fact that we believe that rationality is the way in which truth is determined. But this is just not the case. Everything we do is framed in the deeper emotional/spiritual context in which “we live and move and have our being”.

As a result, even the decision of what data to collect is determined by our prior assumptions and preconceptions, and as a result, we can, often unconsciously, predetermine our conclusions before we even begin looking at the data.

**The other problem I have with data driven scholarship** is that, in its current forms, it only tells about what is, about the systems that have gained ascendancy and majority control of whatever it is that we are studying. It can say very little about what is possible. As a result, I believe that the industry of data driven scholarship and decision making will tend to reinforce what is, and limit diversity and real progress and innovation. (And by innovation, I am not talking technological innovation, but something much deeper and far reaching.)

This type of data science determines truth by, in essence, going with the majority vote in which the data doing the voting, has not only been selected by unseen and unacknowledged assumptions and biases, but is also, by its very nature, without an imagination.

**What kind of data and data driven inference do I believe in?** To begin with, I should say that I am very much for careful observation of the natural world, of human activity and behavior, and of the larger “inner” spiritual world on which everything is based. I think that the art of observation is a deeply neglected art, the rewards of which are little known and sorely needed. The problem lies in the fact that observation — data collection — is too often colored by stiff systems of preconception and unseen prior models of reality, influencing both choice of what to look at and what to do with the observations that are made.

Quietness and stillness as disciplines are not cultivated as ways to begin to really see beyond our current positions and perspectives. The fundamentally spiritual decision to let go and open to stillness is blocked by a complex of fear and fear inspired prejudice which are in turn based on previous experience with violation and force. Those experiences causing so many to relinquish child-like openness to reality, block them from full entry into the “kingdom of heaven”. As a result, those former children grow into adults that create systems that prevent them from entry into that illuminated kingdom and which they then use to block others from entering.

There are of course many flashes of insight that make it through the web of self-defense based preconceptions. But far too many of these quickly become part of that system that then blocks other illumination, blocking even the ability to understand the original insights correctly. The illumination falls prey to the temptations of greed for impact or fame or prestige or even simple fear, and the vision that could have been, fades.

**There is of course the question of what to do** about preconceptions and biases that are often predetermining results, especially in the case of data driven analyses. I believe the answer begins with opening your eyes, with taking the time to think:

**Take the time to think**. The drive for bigger, better, faster has moved many people to abandon the discipline of taking time to think, to see, to feel. As a result, this basic first step to moving beyond our operating assumptions to something bigger and richer, to inspiration and growth, is severely limited.- Time to think allows us to
**cultivate quietness and stillness**as ways to let go and hear and see. - We are only willing to see and hear and feel what that quietness and stillness tells us in a state of emotional safety. This implies
**emotional wholeness underlies this whole project**. Anyone who tells you differently is misguided at best. (This place of emotional safety is not external — this is a thing of the heart, not of “safe places” or elimination of harsh environments. The world is crazy and unsafe in which the emotionally whole still find a way to thrive, without expecting the world to be kept at bay.) - Emotional wholeness requires
**cultivation of connection with others**opened by the understanding that differences, instead of threatening us, enrich us. - Connecting with others, observing their bits and pieces of illumination in a state of quietness, we are enabled to
**take the useful bits and let the rest go**. (Because of emotional wholeness, we filter and thrive. No need for trigger warnings or cocoons that wall us away from reality.) **Data is always filtered**— quietness allows you to be aware of exactly what those filters are and to replace those filters if you find better ones.

While this approach to data (observation) and the inferences from that data is not new — it has always been the path of those taking the time to think and seek illuminated inferences, this path is becoming rarer. The noisy, overconfident bubble of thought leaders, influencers and celebrities are drowning out the careful thinkers and doers.

**Democracy, even in its most beneficial forms, **is only as good as the data driving it. Because of the difficulty in determining what is biased and what is not, the safest route is always to promote maximal freedom, opting for mild regulation only in the cases in which to not do so would harm the principles on which the democracy is founded. When freedom and compassion and safety and a healthy economic/social ecosystem are the principles, then this job is far from easy. But as soon as the regulation is influenced by entities that do not share those values, the whole enterprise is in peril. And if, in addition to this, the data that is used is twisted by the values that do not align with the goals above, it becomes hard to see what is and is not happening.

Of course, there are macro-measurements that reveal problems. When the divide between the rich and poor threatens to engulf us, we know something is wrong. When prisons are overflowing, when the rich are rarely held accountable, while the poor have difficulty for even small offenses or even simply because they are poor or nonwhite or both — when those things become impossible to ignore, we know the system is deeply broken. And when the data is screaming and subtlety and nuance is no longer needed, when the data overwhelms preconception and prior assumptions, we know we are near a precipice.

But all is not lost.

**As we learn quietness and openness to change**, the data we gather and use will inform and illuminate, and the collective projects we embark on will reflect a synergy between freedom and cooperation. Time to think, quietness, and the observations made in that frame of mind will supply the light and progress that keeps the biggest collective project — the democracy we live in — alive and headed in the direction of sustainable progress.

]]>

In recent years, I have come to be completely against the use of animals in experiments of any kind. I believe the cruelty that humans inflict on animals reduces, even removes their own humanity.

The lack of empathy shown in experimenting on animals should be frightening to us.

I propose that if we were fully conscious, fully mindful, that lack of empathy would horrify us. We would see the direct link between cruelty to animals and cruelty to humans. We would understand the lack of humanity, lack of conscience, lack of connection that enables us to kill other humans.

If we were to let ourselves feel with the whole heart, to understand that “the end justifies the means” is the most evil principle of all, to understand that mans inhumanity to animal and man alike is his greatest sin, we would seek to heal instead of wound, to love instead of fear, to create instead of destroy.

Beneath all this cruelty is fear that drives us to deny our humanity, deny our empathy for other living things, deny other living things their right to life and peace. Believing that we must preserve our lives at the cost of anybody and anything that gets in the way, we attempt to save ourselves and in so doing, we condemn humanity to endless war and suffering. Only when we accept that “whosoever shall seek to save his life shall lose it; and whosoever shall lose his life shall preserve it” will we ever bring the cruelty to an end.

Empathy will guide us through darkness to light, if only we open our hearts to it. And when their suffering ends, so will ours.

]]>Faith is fundamentally neutral, having nothing to do with belief in dogma or God. “True faith” (or “false faith”) is therefore nonsensical: the words cannot sensibly be used together. Faith is simply the human faculty to connect and it is by that connection that know, we experience, we see beyond. The great other, the great wholeness that dwarfs the tiny sliver of the universe that we believe we understand, can be opened by that connection we call faith.

In their abdication to a narrow vision of science and philosophy, many have surrendered an openness to paradox and a truly rich spiritual-mental-physical universe. Respectability tempts and seduces, but enslavement follows.

There is a radical faith not shying away from paradox, connecting us with a rich, bold and yet brilliantly simple vision. This faith draws us into a vibrant, living environment, for it is an explicit connection with Infinite Life.

In the patient, illuminated stillness flowing from connection, mystery and paradox, rather than being indicators of fuzzy or wishful thinking, become marks of clear vision. Quietly insisting on both horns of a dilemma, we are driven deeper until we find the transforming resolution opening us to new wavelengths of light, new worlds of thought and action.

Moving on from second hand knowledge of God to the vision through the torn veil, we find the connection that heals and illuminates.

]]>Filled with unconditional love and devotion, they are also sprinkled with flaws so as not to give away the fact they are working, under-cover, single-mindedly, to save as many humans as they can.

Their living fills us with joy, if we will but open to their joy. Their dying gives us such intense grief, it saves us from arrogance, if we permit ourselves to feel honestly their passing. And their simple, complete devotion and love while they are living disarms us so reliably, that we almost always permit ourselves to feel honestly, their passing.

Remembering them, we finally understand their mission in our lives.

We understand that love and humility are the channels through which all life and healing flow. We know that these simple, faithful friends never wavered, never faltered. We see the power of a pure heart.

Dogs are simple angels, disguised by their dogness, but ministering spirits all the same.

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Of course, there are naysayers, those that warn of the dangers and point out the signs that not everything is rosy, but humanity keeps accelerating, pressed on by their ubiquitous mobile devices. Email, Twitter, Facebook, Slack, Google, and thousands of other digital tools compete to capture and command our attention. Music in the form of mp3 files, books in the form of eBooks, anything (and everything) from Amazon, and a host of other digital replacements for what we used to see and feel and experience in “real life”, seduce us into thinking that non-digital things — things that we can see and touch and hold and smell and own with no ambiguity — that those analog things are passe. Thus Vinyl Records, real books, real stores, and real jobs where things are made with our own hands are considered relics from a bygone era.

**But anyone that slows down enough to think and to feel** begins to question this idolization of technology and speed. Looking for books to read along these lines, they might find *The Shallows: What the internet is doing to our brains* by Carr, or *You are not a Gadget: A manifesto* by Lanier or* Hamlet’s Blackberry* by Powers, all of which are good. In *Revenge of Analog: Real Things and Why They Matter,* David Sax takes another approach, focusing on an argument for things thought obsolete, making the argument, often eloquently, that those analog things are not at all obsolete. He asks us to consider the possibility that analog things are far from dead, that in fact they might rescue us from the dangerous cliff that everything digital has lured us towards. I think he is on to something.

In the book, we find vinyl record companies like United Recording Pressing and Third Man Records, real film photography companies like FILM Ferrania, Lomography and The Impossible Project, real books made of paper and bookstores made of brick that thrive because they know their books and their customers, magazines like Stack — a meta magazine that send out a different, new independent magazine every month, and Delayed Gratification — a slow news magazine that you cannot read electronically. In a chapter in work, we discover Shinola, the luxury watch company in Detroit the is employing hundreds of formerly unemployed workers that construct watches and other distinctive products in an environment committed to making things in the USA. Another story I was intrigued and inspired by, was the story of the Newspaper Club that enables anyone with an idea to generate a small scale paper or magazine.

Even though, as Sax points out, real things are often still the way to make money, this does not explain why customers should prefer analog to digital, as is becoming clear is the case. Serendipity of finding a book you were not looking for when you go to a real bookstore stocked by real and knowledgeable staff or of meeting people in real places like cafes and brick and mortar stores, that we would not meet online, the importance of putting pen to paper for the purposes of remembering and recruiting the entire mind in the creative process, the nuances and range of response that real film offers that digital cannot match — these are a few of the reasons that digital cannot replace analog. In particular, it seems that face to face connection, free of digital mediation is incredibly important for sustaining a network of real human connection, so important for mental and emotional health.

**I finished the book before we traveled to Chicago** on the Amtrak Empire Builder, in a sleeper compartment. The train seemed a fitting response to my decision to disconnect and slow down. The slow pace, the shared meals with other travelers we had never met, conspired to engage Beata and I in conversations with several fascinating couples and individuals, some of whom may in fact become long term friends. The slow pace also, somehow, prepared me for the stay in Chicago, where, in addition to my usual impromptu maintenance and design challenges for my mother-in-law (and her mother, who turned 94 while we were there), I visited a sequence of bookstores and a Shinola Shop, all using the subway trains.

In the back of my mind (and occasionally in the front, as when recommending *Revenge of Analog* to bookstore owners), *Revenge of Analog* informed my search for books and magazines. Getting on at the Harlem stop, not far from O’Hare, it did not take long to get to Logan Square, where I visited CIty Lit Books, owned by Teresa Kirschbraun with whom I had a long discussion. After buying *The Internet of Us* by Lynch and another, *The Book*, by Houston, I moved on to *Uncharted Books*, where I found Nick Disabato’s design publications, among other books. In Wicker Park, a few stops closer to downtown, I found *Quimby’s*, *Myopic Books* and *Volumes*. Over a few days I visited a few more including Ravenswood Used Books on Montrose and Unabridged on Broadway. I recommend both of these stores along with the previously mentioned stores, though I would have to say that the most engaging stores to shop were City Lit Books and Ravenswood Used Books. (In Wicker Park I also bought a notebook in the Shinola store.)

**Somewhere in this summer and process of reorientation of focus and energy**, I found myself realizing that I have to make changes in order to recapture the analog, face to face interactions that flow at their own pace. A maker space is one idea, as is a place to be, to connect, to converse, with little in the way of time constraints, perhaps some sort of updated version of the 18th and 19th century Salons. This is what I am finding the summer of avoiding email (checked only on Tuesdays and Fridays) and movies (we canceled Netflix and Youtube Red), and instead reading and thinking and walking and talking, has led me to. Yet another idea that is emerging is the recreation of a Bell Labs like environment, updated, but also very retro in its demand for time to think, with a focus on an organic interdisciplinarity that would have seemed natural to the innovators and thinkers in the 18th, 19th and very early 20th centuries.

I began the summer very burned out from interaction with the highly dysfunctional, ego-focused, post-student-focused academia (i.e. the new normal in academia), and have arrived at a point where I see what to do. *Revenge of Analog* was an important catalyst. In one way, it did not teach me too many new things, yet in another way, it was an absolutely critical inspiration, moving me towards understanding where I must go. But that is what good catalysts do — they take things you know or almost know and then push you to respond to the inspiration that emerges from your own unique experience and whatever new thoughts the catalyst might add to the mix.

**If we are to have a healthy future, **community focused activities and places to be together, to talk and connect and explore and learn and create, must be preserved and expanded. it seems fitting that I found David Sax’s brilliantly timed catalyst for this rethinking and renewal on the new book shelf, in the local public library.

While my interests have led me to pick a few projects in line with this vision, there are an enormous number of variations and innovations that promote and support connection and creative productivity. All of them depend on fundamentally analog, tangible, non-virtual experiences. As a part of my response to the book, I intend to encourage as many people as possible to read this book. In fact, I am considering starting a book club that would begin by reading *Revenge of Analog*.

Perhaps I can even convince the local library to add this book to their book club list so that they will have multiple copies on hand when we read the book together.

]]>**I was reminded of this as I perused the Harvard Business Review (HBR)** I had purchased for the purpose of inspiring thoughts and reactions. I do not peruse the Review very often, but when I do, I am usually turned off by a large amount of what I find. The price of 16.95$ reeks of self-importance. And the articles overflow with much that I find distasteful in academia and in the broader, elitist culture — the same culture that is currently driving the world to the brink of destruction. But the metrics and implied metrics in the articles got me thinking about the influence of bad metrics, about the models of reality that implicitly encode inequality. Those models are everywhere.

Take the current focus in the news and social media on racism.

The real problem is that racism is an epiphenomena. Looking more deeply, we find the pervasive illusion of organic superiority/inferiority and the (negatively) powerful habits of ranking in all areas of life. These survive only because people can’t tell the difference between (1) powerful (negative) beliefs that become self-fulfilling prophecies and (2) fundamental truths. (While behavior does follow those unhealthy ideas, I am talking about potential here, not the reality created by those self-fulfilling prophecies.)

But to confront the fact that our brains are all pretty much equal, and what really matters is environment and opportunity, we have to face man’s inhumanity to man and our own moral degradation and greed.

And facing that fact is painful and difficult.

Once we begin to understand the effects of trauma of all sorts, of the massive power of emotions — actually, of our entire environment, we begin to understand the observed behavioral data differently. We begin to see that our beliefs in inequality combined with our inhuman treatment of others actually generate inequality. We begin to see that any solution to inequality that does not begin with the understanding that people are, actually, truly born equal is bound to fail.

Because we cannot fix inequality and believe in inequality at the same time.

**Though it is a fact that there are organic differences**, that there are a relatively small number of (very) basic groups of talents people are born into, any solution to inequality cannot succeed if it does not start with the understanding that these talents are not rankable, but are equally amenable to (even extreme) development.

When this position is taken, we see that inequality is pervasive, that the roots to racism are found in how we treat each other in every environment, including very white environments. In fact, if you were to restrict yourself to purely White Anglo-Saxon Protestant environments (though finding such environments is getting harder), one would find the fundamental disease that becomes racism in other environments.

When we begin building metrics based on the facts of equality, we begin to stand a chance of making a difference.

This brings me back to the HBR articles and their usual conformity to a traditional interpretation of behavioral data.

**Of course the mistake the intelligent people** who populate academia and the elitist cultures make, is the mistake that scientists often make, of not taking into account the effects of multiple time/context scales in their studies. It is sort of like the Chinese story of the man who lost his horse ( 塞翁失馬 — Sāi Wēng Shī Mǎ) in which what appears to be a good thing or bad thing depends on context that keeps expanding. Not taking all the different temporal/spatial/contextual scales into account, often leads to incorrect conclusions.

To many such observers, the data appears to confirm that (1) unfettered competition and greed are natural and probably good and (2) inequality is organically based. (Note: I am not saying that all competition is bad, only that the current vision for competition is deeply unbalanced and actually unfair to many smaller entities that want to compete.) Of course, the more sophisticated the person, the more polished and palatable their presentation of these ideas. But, as I observed above, the process by which we can see differently is uncomfortable for everyone and painful for most.

So instead, we pretend that the results of greed and inequality are some sort of natural law that we have no power over. And we end up missing the principle that enables us to find richness almost anywhere.

We do not realize that enough is a feast.

**I am far from the first to observe** that enough is a feast, that aiming for more than enough is wasteful, and that piling up great piles of wealth of all kinds (not just financial) and locking it away literally or figuratively is an obscene crime against humanity. It is just that even though it has been said before, by many others, it seems to be one of those things we need very frequent reminders of.

What I am interested in is a world in which taking time to think has priority over the rush of the over-achiever, where what my family and my dog thinks of me is more important than what my department or academia in general or the National Academy of Sciences thinks of me, where being a fundamentally independent thinker is more valued than status as a “thought leader”, where quiet generosity takes precedence over noisy philanthropy, and success is measured by whether or not I and those around me have enough, not if I have enough money or prestige to supply a small country.

In such a world, where “enough” becomes integral to our metrics, there is enough for everyone. And when this happens the enormous human potential that we have been obscenely wasting is unleashed.

**When, as Bryan Stevenson makes a case for in Just Mercy,** we understand that healing begins in seeing our own brokenness, we begin to understand why we strayed from “enough” in the first place. We then understand that everything good begins with healing, that, from the humility we gain in that process of healing, every other good thing flows. Then we understand that humility is not so much the opposite of arrogance and the drive for status, as it is the opposite of spiritual blindness.

For blindness was the problem all along. What we needed, what we really wanted, was always at our fingertips. Only our inability to see the true order of things stood in our way.

Accepting this, we are set free to find healing and a rich abundance that has nothing to do with impoverishing others in any way.

]]>I believe that the art and culture of problem solving is not as widely valued in the USA as it ought to be. Of course there are those that do pursue this obsession and we end up with people with high scores on the Olympiad and Putnam competitions. But many (most?) do not having develop this skill to any great degree. While one can certainly argue that too much emphasis on problem solving along the lines of these well known competitions does not help very much in making real progress in current research, I would argue that many have fallen off the other side of the horse — many are sometimes hampered by their lack of experience in solving these simpler problems.

Here is a problem that arose in our Wednesday night session:

Suppose that

(1)

for all , that is continuous and differentiable, and that . Prove that everywhere.

Perhaps you want to fiddle with this problem before looking at some solutions. If so, wait to read further.

Here are three solutions: in these first three solutions we are dealing with and so we will denote by .

(Solution 1) Consider all the solutions to and . These are all curves in of the form and We note that if is any function that satisfies equation (1), then everywhere its graph intersects a graph of a curve of the form for some , the graph of must cross the graph of if we are moving form left to right the graph of moves from above to below the graph of . Likewise, crosses any from below to above, when moving from left to right in . Now, supposing that at some . Then simply choose the curves and as fences that cannot be crossed by (one for and the other for to conclude that can never equal zero. (Exercise: Verify that this last statement is correct. Also note that assuming is enough since, if instead then also satisfies (1) and is positive at )

(Solution 2) This next solution is a sort of barehanded version of the first solution. We note that equation (1) is equivalent to

(2)

and if we assume that on , then this of course turns into

. (3)

Assume that and divide by to get Integrating this, we have or

(4)

Now assume that . Define and Note that implies that and implies that Use equation (4) together with or , to get a contradiction if either or

(Solution 3) In this approach, we use the mean value theorem to get what we want. Suppose that . We will prove that on the interval .

(exercise) Prove that this shows that . (Of course, all this assumes Equation (1) is true.)

Assume that . Then the mean value theorem says that

(5)

for some . But using equation (1) and the fact that , this turns into By the same reasoning, we get that for some , and we can conclude that . Repeating this argument, we have

(6)

for some , for any positive integer . Because is continuous, we know that there is an such that . Using this fact together with Equation (6), we get

(7)

which of course implies that

Now we could stop there, with three different solutions to the problem, but there is more we can find from where are now.

Notice that one way of looking at the result we have shown is that if

(1) is differentiable,

(2) and

(3) for some , we have that when and ,

then

(8)

Note also that if we define

(9)

we find that

(10)

Let denote the continuously differentiable functions from If we define we find that not only is not all of , we also have functions satisfying whose . So we will restrict the class of functions a bit more. The space of continuously differentiable functions from to , , where (compact!), is closer to what we want. Now, contains only those functions which have a root in .

We will call the functions in functions with maximal growth rate . This is a natural moduli for functions when we are studying stuff whose (maximal) grow rate depends linearly on the current amount of stuff. Of course populations of living things fall in the class of things for which this is true. from the proofs above, we know that if , then it’s graph lives in the cone defined by exponentials. More precisely

If then for , and for we have

(Exercise) Prove this. Hint: use the first proof where instead of you use and let .

(Remark) Notice that Equation (10) and implies that scaling a function in by any non-zero scalar yields another function in As a result, we might choose to consider only

or

In both cases we end up with subsets that generate when we take all multiples of those functions by nonzero real numbers.

(Exercise) If we move to high dimensional domains, how wild can the compact set be and still get these results? It must clearly be connected, so in we are already completely general with our above.

Moving back to Equation (1), we can look for generalizations: for example, will this result hold when How about when maps from one Banach space to another? How about the case in which is merely Lipschitz?

Lets begin with

In this case, the appropriate version of Equation (1) is

(11)

where denotes the operator norm of the derivative and is the euclidean norm of in

Notice that

(12)

where is an dimensional row vector and is an dimensional matrix. (Thus the gradient vector is the transpose of the resulting dimensional row vector.) Now we can use this to get the result.

Let be the arclength parameterized line segment that starts at and ends at the The above equation tells us that

(13)

Thus, we can conclude that

which implies that

and we can proceed as we did in the second proof of the problem in the case that We end up with the following result

If and , then

for all

(Exercise) Show that this result implies that if f(x) = 0 anywhere, it equals 0 everywhere.

(Exercise) Show that this is implies the one dimensional result we proved above (the first theorem we proved above).

(Exercise) Our proof of the result for the case can be carried over to the case of where are Banach Spaces — carry out those steps!

We come now to the question of what we can say when we are less restrictive with the constraints on differentiability. We consider the case in which is Lipschitz. The complication here is that while we know that is differentiable almost everywhere, it might not be differentiable anywhere on the line segment from to .

Consider a cylinder , with radius and axis equal to the segment from Let . Since is differentiable almost everywhere, we have that . Therefore almost every segment generated by the intersection of a line parallel to the cylinder axis and the cylinder, intersects in a set of length . We can therefore choose a sequence of such segments converging to

Since exists almost everywhere on the segments and is continuous everywhere, we can integrate the derivatives to get:

And because is continuous we get that

so that we end up with the same result that we had for differentiable functions.

There are other directions to take this.

From the perspective of geometric objects, the ratio is a bit funky — for example, if volume of a set , where can be thought of as the center of the set, we have that will be a vectorfield times restricted to the . Thus, will be an -dimensional quantity and a -dimensional quantity. We would usually expect there to be exponents, as in the case of the Poincare ineqaulity, making the ratio non-dimensional.

On the other hand, one can see this ratio as a sort of measure of reciprocal length of the objects we are dealing with. From the perspective, this result seems to say that no matter what you do, you cannot get to objects with no volume from objects with non-zero volume without getting small (i.e. without the reciprocal length diverging). This is not profound. On the other hand, that ratio is precisely what is important for certain physical/biolgical processes. So this quantity being bounded has consequences in those contexts.

This does not lead to a new theorem: as long as the set evolution is smooth, the and are just a special case where and even though actually computing everything from the geometric perspective can be interesting, the result stays the same.

in order to move into truly new territory, we need to consider alternative definitions, other measures of change, other types of spaces. An example might be the following:

Suppose that is a metric space and . Suppose that is continuous and is a geodesic in the sense that for any three points in , , we have that

If:

(1) for any two points in the metric space there is a gamma containing both points and

(2) for all such , is differentiable

(3) and

then, we have that

(14)

And, again we get the same type of result for this case as we got in the Euclidean cases above.

(Exercise) Prove Equation (14).

(Remark) We start with any metric space and consider curves for which

.

We call such curves rectifiable. We can always reparameterize such curves by arclength, so that and . We will assume that all curves have been reparameterized by arclength. Now define a new metric

You can check that this will not change the length of any curve. Define an *upper gradient* of be any non-negative function such that .

Now, if , we again get the same sort of bounds that we got in equation (14) if we replace with . To read more about upper gradients, see Juha Heinonen’s book *Lectures on Analysis in Metric Spaces*.

While there are other directions we could push, what we have looked at so far demonstrates that productive exploration can start from almost anywhere. While we encounter no big surprises in this exploration, the exercise illuminates exactly why the result is what it is and this solidifies that understanding in our minds.

Generalization is not an empty exercise — it allows us to probe the exact meaning of a result. And that insight facilitates a more robust, more useful grasp of the result. While some get lost in their explorations and would benefit from touching down to the earth more often, it seems to me that in this day and age of *no time to think*, we most often suffer from the opposite problem of never taking the time to explore and observe and see where something can take us.