# Dual Tyrannies of Data and Democracy (and what to do about it)

In this new age of extremes, celebrity and elitism without bounds, those that pride themselves on their enlightenment often make a big deal about being democratic in their ambitions and data driven in their thinking and reasoning.

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:

1. 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.
2. Time to think allows us to cultivate quietness and stillness as ways to let go and hear and see.
3. 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.)
4. Emotional wholeness requires cultivation of connection with others opened by the understanding that differences, instead of threatening us, enrich us.
5. 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.)
6. 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.

# Freedom and Writing

1. Communicate in person often, one on one, one on few, spontaneously (even if you have prepared for hours).
2. Write papers and books that you are inspired to write … if you stick to true inspiration, there might not be that many papers and books, but what there is will be very good.
3. Write ideas compulsively, write, write, write … but do not feel the necessity of publishing most of it, even if you make most or all of it available as notes, online.
4. See publication, or even posting online, less as carving something into stone and more as an invitation for others to join in your explorations.
5. Never let the writing replace thinking — take the time to think (and plenty of it).
6.  Don’t be afraid of mistakes and don’t edit as you write — edit by iteration, by revision.
7. Polish your writing guided by one simple rule — to make your thoughts clearly visible to your readers. Forget all the other rules for proper writing, for how to write.
8. Creativity is heightened by simple playfulness. Cultivate it.
10. Sustain your inspiration by a commitment to freedom (with kindness) + connection (with generosity).

# Faith Is Connection

Some use faith to connect with finite, comfortable places, perhaps in response to fears of one kind or another. Little boldness is required. Others courageously connect to the ragged edge, but without the calm quietness that warns of errors, omissions or even danger. Yet others refuse to acknowledge the fundamental, critical role of faith, making the mistake of identifying this primitive human faculty with something that implies a belief in God.

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.

# Metrics and Inequality

Metrics — measures of performance or value — drive what we do at every scale, from the small, individual scale to the massive global scales. When those metrics are founded on misconceptions of reality, they contort behavior in such a way as to appear to support those misconceptions. To get back to the natural order of things, away from the artificial reality created by those false beliefs, we must start by reseting our metrics.

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.

# Fun with simple analysis problems I

This last semester, I ran a fun, informal master class in problem solving. Actually, a graduate student of mine — Yunfeng Hu — who is an expert problem solver, produced all the problems from the immense library he built up over his undergraduate career in China.

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

$|\frac{df}{dx}(x)| \leq \lambda |f(x)|$                    (1)

for all $x$, that $f$ is continuous and differentiable, and that $f(0) = 0$. Prove that $f(x) = 0$ 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 $f:\Bbb{R}\rightarrow\Bbb{R}$ and so we will denote $\frac{df}{dx}$ by $f'$.

(Solution 1) Consider all the solutions to $g' = 2\lambda g$ and $h' = -2\lambda h$. These are all curves in $\Bbb{R}^2$ of the form $y = g_{C}(x) = Ce^{2\lambda x}$ and $y = h_{C}(x) = Ce^{2\lambda x}.$ We note that if $y=f(x)$ is any function that satisfies equation (1), then everywhere its graph intersects a graph of a curve of the form $y = g_{C}(x) = Ce^{2\lambda x}$ for some $C\in\Bbb{R}$, the graph of $f$ must cross the graph of $g_{C}.$ if we are moving form left to right the graph of $f$ moves from above to below the graph of $g_{C}$. Likewise, $f$ crosses any $h_{C}$ from below to above, when moving from left to right in $x$. Now, supposing that $f(x*) > 0$ at some $x*$. Then simply choose the curves $g_{\frac{f(x*)}{e^{2\lambda x*}}}(x)$ and $h_{\frac{f(x*)}{e^{-2\lambda x*}}}(x)$ as fences that cannot be crossed by $f(x)$ (one for $x < x*$ and the other for $x > x*$ to conclude that $f(x)$ can never equal zero. (Exercise: Verify that this last statement is correct. Also note that assuming $f(x*) > 0$ is enough since, if instead $f(x*) < 0$ then $-f$ also satisfies (1) and is positive at $x*$)

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

$-\lambda |f(x)| \leq f'(x) \leq \lambda |f(x)|$            (2)

and if we assume that $f(x) > 0$ on $E\subset\Bbb{R}$, then this of course turns into

$-\lambda f(x) \leq f'(x) \leq \lambda f(x)$.            (3)

Assume that $[x_0,x_1]\in E$ and divide by $f(x)$  to get  $-\lambda \leq \frac{f'(x)}{f(x)} \leq \lambda.$ Integrating this, we have $-\lambda (x_1 - x_0) \leq \ln(\frac{f(x_1)}{f(x_0)}) \leq \lambda (x_1 - x_0)$ or

$e^{-\lambda (x_1 - x_0)} \leq \frac{f(x_1)}{f(x_0)} \leq e^{\lambda (x_1 - x_0)}.$            (4)

Now assume that $f(x*) > 0$. Define $u = \sup \{ w | f(x) > 0 \text{ for all } x* < x < w\}$ and $l = \inf \{ w | f(x) > 0 \text{ for all }x* > x > w \}.$ Note that $l \neq -\infty$ implies that $f(l) = 0$ and $u \neq \infty$ implies that $f(u) = 0.$ Use equation (4) together with $\{\text{a sequence of }x_0\text{'s} \downarrow l \text{ and }x_1 = x*\}$ or $\{x_0 = x* \text{ and a sequence of }x_1\text{'s} \uparrow u\}$,  to get a contradiction if either $l \neq -\infty$ or $u \neq \infty.$

(Solution 3) In this approach, we use the mean value theorem to get what we want. Suppose that $f(x_0) = 0$. We will prove that $f(x) = 0$ on the interval $I = [x_0 - \frac{1}{2\lambda}, x_0 + \frac{1}{2\lambda}]$.

(exercise) Prove that this shows that $\{f(x) = 0\text{ for some } x\}$ $\Rightarrow$ $\{f = 0\text{ for all } x\in\Bbb{R}\}$. (Of course, all this assumes Equation (1) is true.)

Assume that $x\in I$. Then the mean value theorem says that

$|f(x) - f(x_0)| \leq |f'(y_1)| |x - x_0| \leq |f'(y_1)| \frac{1}{2\lambda}$               (5)

for some $y_1\in I$. But using equation (1) and the fact that $f(x_0) = 0$, this turns into $|f(x)| \leq \frac{1}{2}f(y_1).$ By the same reasoning, we get that  $|f(y_1)| \leq \frac{1}{2}f(y_2)$ for some $y_2 \in I$, and we can conclude that $|f(x)| \leq \frac{1}{2^{2}}f(y_2)$. Repeating this argument, we have

$|f(x)| \leq \frac{1}{2^{n}}f(y_n)$                  (6)

for some $y_n \in I$, for any positive integer $n$. Because $f$ is continuous, we know that there is an $M < \infty$ such that $f(x) < M \text{ for all } x\in I$. Using this fact together with Equation (6), we get

$|f(x)| \leq \frac{M}{2^{n}} \text{ for all positive integers } n$                (7)

which of course implies that $f(x) = 0$

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) $f$ is differentiable,

(2) $f(x_0)=0$ and

(3) for some $\delta > 0$, we have that $f(x) \neq 0$ when $x \neq x_0$ and $x\in [x_0 - \delta, x_0 + \delta]$,

then

$\limsup_{x\rightarrow x_0}A_{f}(x) \equiv \left|\frac{f'(x)}{f(x)}\right|\rightarrow\infty$              (8)

Note also that if we define

$a(f) \equiv \sup_{x\in\Bbb{R}} A_{f}(x)$                  (9)

we find that

$a(f) = a(\alpha f) \text{ for all }\alpha\neq 0.$               (10)

Let $C^{1}(\Bbb{R},\Bbb{R})$ denote the continuously differentiable functions from $\Bbb{R}\text{ to }\Bbb{R}.$ If we define $C_{\lambda} = \{f | a(f) \leq \lambda\}$ we find that not only is $\bigcup_{n\in\Bbb{Z}^{+}} C_n$ not all of $C^{1}(\Bbb{R},\Bbb{R})$, we also have functions satisfying $0 < b \leq f(x) \leq B < \infty$ whose $a(f) = \infty$. So we will restrict the class of functions a bit more. The space of continuously differentiable functions from $K\subset \Bbb{R}$ to $\Bbb{R}$, $C^{1}(K,\Bbb{R})$, where $K = [-R,R]$ (compact!), is closer to what we want. Now, $C_{\infty} \setminus\bigcup_{n\in\Bbb{Z}^{+}} C_n$ contains only those functions which have a root in $K$.

We will call the functions in $C_{\lambda} \subset C^{1}(K,\Bbb{R})$ functions with maximal growth rate $\lambda$. 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 $f\in C_\lambda$, then it’s graph lives in the cone defined by exponentials. More precisely

If $a(f) = \lambda$ then for $x < x_0$,   $\frac{f(x_0)}{e^{\lambda x_0)}}e^{\lambda x} \leq f(x) \leq \frac{f(x_0)}{e^{-\lambda x_0)}}e^{-\lambda x}$   and for $x > x_0$ we have $\frac{f(x_0)}{e^{-\lambda x_0)}}e^{-\lambda x} \leq f(x) \leq \frac{f(x_0)}{e^{\lambda x_0)}}e^{\lambda x}.$

(Exercise) Prove this. Hint: use the first proof where instead of $2\lambda$ you use $\alpha\lambda$ and let $\alpha\downarrow 1$.

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

$F \equiv f\in c_\lambda\text{ such that }f(0) = 1$

or

$F \equiv \{\text{ functions whose minimum value on }K\text{ is }1\}.$

In both cases we end up with subsets that generate $C_\lambda$ 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 $K$ be and still get these results? It must clearly be connected, so in $\Bbb{R}^1$ we are already completely general with our $K$ above.

Moving back to Equation (1), we can look for generalizations: for example, will this result hold when $f:\Bbb{R}^{n} \rightarrow \Bbb{R}^{m}?$ How about when $f$ maps from one Banach space to another? How about the case in which $f$ is merely Lipschitz?

Lets begin with $f:\Bbb{R}^{n} \rightarrow \Bbb{R}^{m}.$

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

$||Df(x)|| \leq \lambda ||f(x)||$                    (11)

where $||Df(x)||$ denotes the operator norm of the derivative $Df(x)$ and $||f(x)||$ is the euclidean norm of $f(x)$ in $\Bbb{R}^m.$

Notice that

$D\ln(||f(x)||) = \frac{1}{||f(x)||}\left(\frac{f(x)}{||f(x)||}\right)^{t}Df(x)$                    (12)

where $\left(\frac{f(x)}{||f(x)||}\right)^{t}$ is an $m$ dimensional row vector and $Df(x)$ is an $n\text{ by }m$ dimensional matrix. (Thus the gradient vector is the transpose of the resulting $n$ dimensional row vector.)  Now we can use this to get the result.

Let $\gamma(s)$ be the arclength parameterized line segment that starts at $x_0$ and ends at $x_1$ the The above equation tells us that

$\int_{\gamma} D\ln(||f(x(s))||) ds = \int_{\gamma} \frac{1}{||f(x)||}\left(\frac{f(x)}{||f(x)||}\right)^{t}Df(x) \leq \int_{\gamma} \frac{||Df(x(s))||}{||f(x))||} ds.$        (13)

Thus, we can conclude that

$\ln(||f(x_1)||) - \ln(||f(x_0)||) \leq \lambda ||x_1 - x_0||$

which implies that

$-\lambda ||x_1 - x_2|| \leq \ln\left(\frac{||f(x_1)||}{||f(x_0)||}\right) \leq \lambda ||x_1 - x_0||$

and we can proceed as we did in the second proof of the problem in the case that $f:\Bbb{R}\rightarrow \Bbb{R}.$ We end up with the following result

If $||Df(x)|| \leq \lambda ||f(x)||$  and $||f(x)|| \neq 0 \text{ for all } x\in B(x*,r)\subset\Bbb{R}^n$, then

$e^{-\lambda ||x - x*||} \leq \frac{||f(x)||}{||f(x*)||} \leq e^{\lambda ||x - x*||}$

for all $x\in B(x*,r).$

(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 $f:\Bbb{R}^n\rightarrow\Bbb{R}^m$ can be carried over to the case of $f:B_1 \rightarrow B_2$ where $B_1\text{ and }B_2$ 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 $f:\Bbb{R}^n\rightarrow\Bbb{R}^m$ is Lipschitz. The complication here is that while we know that $f$ is differentiable almost everywhere, it might not be differentiable anywhere on the line segment from $x_0$ to $x_1$.

Consider a cylinder $C_{x_0}^{x_1}(1)$, with radius $1$ and axis equal to the segment from $x_0\text{ to }x_1.$ Let $E = C_{x_0}^{x_1}(1) \cap \{x| Df(x)\text{ exists }\}$. Since $f$ is differentiable almost everywhere, we have that $\mathcal{L}^n( C_{x_0}^{x_1}(1)\setminus E) = 0$. Therefore almost every segment $L$ generated by the intersection of a line parallel to the cylinder axis and the cylinder, intersects $E$ in a set of length $||x_1 - x_0||$. We can therefore choose a sequence of such segments converging to $[x_0,x_1].$

Since $Df$ exists $\mathcal{H}^1$ almost everywhere on the segments $[x_0^k, x_1^k]$  and $f$ is continuous everywhere, we can integrate the derivatives to get:

$-\lambda ||x_1^k - x_0^k|| \leq \ln\left(\frac{||f(x_1^k)||}{||f(x_0^k)||}\right) \leq \lambda ||x_1^k - x_0^k||.$

And because $f$ is continuous we get that

$-\lambda ||x_1 - x_0|| \leq \ln\left(\frac{||f(x_1)||}{||f(x_0)||}\right) \leq \lambda ||x_1- x_0||.$

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 $\frac{||Df||}{||f||}$ is a bit funky — for example, if $f(x) =$ volume of a set $E(x)\subset \Bbb{R}^n = \mathcal{L}^n(E(x))$, where $x$ can be thought of as the center of the set, we have that $Df$ will be a vectorfield $\eta$ times $\mathcal{H}^{n-1}$ restricted to the $\partial E(x)$. Thus, $||Df||$ will be an $n-1$-dimensional quantity and $f$ a $n$-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 $f$ and $Df$ are just a special case where $f:\Bbb{R}^n\rightarrow\Bbb{R}^1$ 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 $X$ is a metric space and $f:X\rightarrow \Bbb{R}$. Suppose that $\gamma:\Bbb{R}\rightarrow X$ is continuous and is a geodesic in the sense that for any three points in $\Bbb{R}$, $s_1 < s_2 < s_3$, we have that $\rho(\gamma(s_1),\gamma(s_3)) = \rho(\gamma(s_1),\gamma(s_2)) + \rho(\gamma(s_2),\gamma(s_3)).$

If:

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

(2) for all such $\gamma$, $g_{\gamma} \equiv f\circ\gamma$ is differentiable

(3) and $\frac{|g_{\gamma}(s)|}{|f(\gamma(s))|} \leq \lambda$

then, we have that

$-\lambda \rho(x_1, x_0) \leq \ln\left(\frac{|f(x_1)|}{|f(x_0)|}\right) \leq \lambda \rho(x_1,x_0).$                       (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 $\gamma:[a,b]\subset\Bbb{R}\rightarrow X$ for which

$l(\gamma)\equiv\sup_{\{\{s_i\}_{i=1}^{n}| a = s_1 \leq s_2 \leq ... \leq s_n = b\}} \sum_{i=1}^{n-1} \rho(\gamma(s_{i}),\gamma(s_{i+1})) \leq \infty$.

We call such curves rectifiable. We can always reparameterize such curves by arclength, so that $\gamma(s) = \gamma(s(t)), t\in[0,l(\gamma)]$ and $l([\gamma(s(d)),\gamma(s(c))] ) = d-c$. We will assume that all curves have been reparameterized by arclength. Now define a new metric

$\tilde{\rho}(x,y) = \inf_{\{\gamma | \gamma(a) = x\text{ and }\gamma(b) = y\}} l(\gamma).$

You can check that this will not change the length of any curve. Define an upper gradient of $f:X\rightarrow \Bbb{R}$ be any non-negative function $\eta_f:X\rightarrow \Bbb{R}$ such that $|f(y) - f(x)| \leq \int_{\gamma} \eta_f(\gamma(t)) dt$.

Now, if $\frac{|\eta_f(x)|}{|f(x)|} \leq \lambda$, we again get the same sort of bounds that we got in equation (14) if we replace $\rho$ with $\tilde{\rho}$. 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.

# Finding Quietness

Rereading parts of Glynne Robinson Betts’ 1981 book, Writers in Residence, recalled simpler, deeper times, when finding places of quietness and taking time to think was part of the routine many people used in order to hear themselves and others. In fact, reading this again prompted me to expand the time I spend without Internet interruptions. Steps as simple as ignoring email for extended periods or as comprehensive as turning the computer off for the entire weekend, are emerging as a necessary part of reclaiming quietness and time to think.

There is nothing profound in these decisions to disconnect — whatever is profound happens as a result of taking that time to see and listen and think.

When I do slow down, every pause, every quietness, every moment taken to see, to listen, to think, rewards with a rich, living connectedness and depth that cannot be exhausted. The fabrics of the past and future join with the present, without seams, without a sense that I am working to recall, to see, to feel. Time opens up, I enter, to travel my own path, to sit or stand or walk … stopping time, finding passage to places beyond space and time.

To the strictly modern intellect, what I have just said probably seems like non-sense. Reason, based on easily observable facts, will find little irrefutable evidence that a skeptic would find compelling.

I therefore offer no argument to convince the skeptic. Instead I say, “Come and see”.

When we begin to let go of dogma, the regard of peers, and the comfort of the in-group, room for discovery is created. Launching into quiet spaces, where fear is replaced by stillness, a boundless infinity surprises. We find flow.  In this personal place without limits, I find an overflowing garden, teeming with life. On the living path, everything is illuminated.

Yet this is something I cannot really transmit. It is only something I can hint at in what I write, faintly, incompletely. The experience of discovery, of knowing, of traveling to those places that are here and beyond at the same time, cannot be captured in words.

To see, you must see though your own eyes. To see, you must choose to slow down, find quietness, and dwell there.

I believe that most – possibly all – human beings have, at one time or another, experienced immersion in flow and a connection to the place without limits. There is a resonance emerging from any such experience, no matter how brief, that enables those with that experience to hear each other.  But life often seems to conspire to crush those memories, to remove our ability to hear and see. In the quiet, we can be moved to remember, to see, to hear. In the quiet we remember the place without limits.

In writing something of what I see and hear, there is a chance that faint recollections will be stirred in those that read, in the way Writers in Residence stirred my memories, my recollections of a time when quietness and time to think was plentiful.

The thought of this possibility brings a subtle sense of connection, of silent conversation, with those as yet undiscovered friends. Lingering in rediscovered quietness, we move against the flow of noise and commotion and modern distraction, encouraging all those in our circle of influence to rediscover for themselves their own place without limits.

# Doing Mathematics

I have come to question a significant portion of the culture in academia, even while I have developed a deeper connection with other parts of that same culture or at least the culture that we could have. While I am deeply committed to mathematics as a creative occupation, and to teaching and mentoring in mathematics, my experience in academia after re-entering it seven years ago has strengthened my rejection of the many parts of that culture because they hinder the best research and teaching.

There are many aspects I could discuss, but here I am singling out four: the question of what makes a mathematical result or paper worthy of recognition together with the place of exposition in mathematics,  the value of awards and recognitions in mathematics, and the effects of federal funding on mathematics and academia.

As opposed to trying to do some sort of statistical study — a study which would only be meaningful if there were sufficient numbers of people following the ideas I propose, and there is not! — I will invoke common sense and intuitions that are commonly agreed on, but usually discarded as a guide for actions because of the economic realities of higher education; the institutions that pay us expect and reward the defective model and very few actively step outside those bounds.

What comes from the idea that results are best if they are definitive? Frankly speaking, I believe this idea is part of a cluster of ideas that impoverishes mathematics and mathematical culture.

I first thought about this when reading Bill Thurston’s 1994 article On Proof and Progress in Mathematics. In this article he contrasted how he approached his first work on foliations (resolve all questions, definitively!) versus his later work in geometry and the huge difference a more generous approach made in creating a rich, open, inspiring environment that many others got involved in, rather than the pinnacle of achievement that was admired from a distance.

Instead of maintaining a museum of monuments, we should propagate a countryside filled with rich, diverse gardens of ideas and a zoo of people tending and changing and expanding and creating new gardens.  While the first model leaves a trail of impressive facts, fit for admiration and worship, the second model is defined by engagement and inspiration for widespread creativity.

When Henry Helson visited Poland after the war, he was struck by the purity and simplicity of the mathematical culture that was also very generous. As he relates in his 1997 Notices article, Mathematics in Poland after the War, he was struck by the combination of generosity and fun that pervaded a culture that was serious about mathematics, but happy to publish things that did not aim to grab and own whole swaths of mathematical territory. Rather they published relatively short papers, each of which presented one new idea very clearly.

That exposition has been neglected, in spite of all the lip service to the contrary, can be seen in the response to the astrobites.org site, which has gained a lot of attention in the astrophysics community because of the large contrast between the high quality exposition that astrobites.org offers and the usual difficulty that non-experts have in reading scholarly papers.

I am now convinced that the high art of exposition should be valued as highly as the construction of brand new theorems, that publishing in such a way as to leave much to others is better than cleaning up an area and creating a monument: that what gets considered valuable mathematics ought to be greatly broadened. If anyone finds value — maybe because of explanations that require original thought, maybe because it brings the ideas to new audiences, maybe because it helps students see something clearly, maybe because it brings the understanding to the general public, and yes, possibly because it is completely original and surprising in construction — then it is valuable mathematics, worthy of the deepest respect. In this new model, the quality of the writing becomes very important. (I suspect that some will take issue with that statement saying that this is not a new model, but I will disagree and point to the enormous quantity of poorly written articles and books, some of which are also very valuable, even though they are not written very well. Of course, there are papers and books that are very, very well written. But it seems that this is considered a cherry on top, rather than something that should always, before anything else, be there.)

I am not urging that there be an effort to police exposition, but rather that this be given a great deal more attention at every level of education and practice. If we must have awards, let them go to those that have explained things well, have written things well. Better yet, train students to pursue the intrinsic rewards of doing anything well, from explaining derivatives to a confused calculus student to proving some new, highly technical theorem.

To encourage such changes, we would need to revisit how we reward and support the mathematical enterprise. This brings us to the consideration of the last two cultural components I said I was going to discuss: awards and federal funding.

Why do mathematics? For me, it is another form of art and at the same time, an exploration of the universe we live in. Knowing and understanding and explaining and inspiring others to do the same, exercises deep creativity and generosity; this is an occupation worthy of human beings that value themselves and others. Of course, there are an enormous number of occupations that can beneficially occupy the human mind and spirit. And each one can be as satisfying and beautiful and useful in its pursuit. By useful, I mean useful as an occupation, not useful as a tool to bend the world to my will. It is the occupation itself that is valuable. What happens to us and those we teach and share with, when we occupy ourselves (in a healthy environment!) is the greatest justification for any occupation.

From this position it becomes clear that awards and honors that many aspire to are actually a distraction. The reward is in the occupation itself. There are of course honors that have more to do with real appreciation rather than ranking and fame, and for such honors there is a place in a healthy culture. But the greed that masquerades in all of us as something more beautiful, seeks fame and fortune as a substitute for love and respect, whose lack actually gives room to that greed in the first place.

When the American Mathematical Society proposed the status of Fellow of the society, the negative side effects of such a program were pointed out rather eloquently by multiple individuals. In particular, I remember that Frank Morgan’s argument against the establishment of the program, and Neal Koblitz’ refusal of the offer of the status of Fellow. Of course, there is also the curious case of Perelman who refused the Fields Medal, the mathematical equivalent of the Nobel prize, whose recipients are given a demi-god status. For an interesting telling of the story and more, see Sylvia Nasar and David Gruber’s article Manifold Destiny in the August 28, 2006 issue of the New Yorker. (In the story, they quote Gromov, another prominent mathematician. Even though I very much doubt Gromov’s explanation of Perelmans refusal as a result of some great purity on Perelmans part, it is a story worth reading and thinking about.)

The influence of federal funding in mathematics, while it has enabled a great expansion of the enterprise, has led to a degradation of the culture, and not only in mathematics. It is well known that federal funding has turned academia into a serious addict, willing to do anything for the next fix of federal funds. That, combined with, spurred on by, the neglect of higher education in the public sector, has led to the very bad state of affairs in which grant money reigns supreme, fame (which can be turned into money!) comes second and teaching, for all the lip service it is given, occupies the lowest realms of academia. Proof of this diagnosis is not needed by anyone in academia (other than administrators who profit from illusions proposing some other reality), but if proof is needed, one need not look any further than the way adjuncts and instructors, who do a great deal of the teaching, are treated. Both in terms of the dismal pay and the insecurity of their jobs, we are saying that teaching is not what a university is really about — it is just what we have to do to keep up the charade.

But this is also where the tragedy lies; it lies in the immense impoverishment that results when teaching is not given top priority. It is a law of nature that real greatness, true stature, is proportional to the service to others that an entity or person actually provides. You may prefer to see this as my definition of greatness and stature. Either way, assuming this to be true, we have traded real nobility for a meager, greedy existence when we accept the perverted system of values that we currently have at research universities — and even, in some ways at teaching universities.

While small liberal arts college do in fact value teaching, they still take advantage of the situation generated by research universities and often pay their adjuncts obscenely low wages. It is tragic and funny at the same time that such colleges are usually full of people who think that businesses ought to raise the minimum wage, provide health care and longer paid vacations, and all sorts of other good ideas, but when it comes to the situation they have power over, they turn a curiously blind eye. But there is also this idolization of research universities, of elite institutions and this admiration pulls in some of the poison that they could otherwise easily avoid.

But, as I wrote in the previous post in this blog,  Learning to Think and to Act, research is a critical piece in education. It inspires and illuminates and brings a freshness and vitality that should be insisted on. On the other hand, research without teaching becomes selfish and elitist and aimed at goals that can at times be silly and irrelevant in their isolation.

What then, can we do? If the system is so far astray, what can be done?

In my opinion, the most powerful thing you can do is inspire change in your own sphere of influence by a focus on the place of freedom you actually have. Having your principles and philosophy aligned with life and love, and consistently acting in accordance with them, has always been the most powerful thing anyone could do.

Creative exploration and teaching, with a deep sensitivity for those that struggle; the pursuit of both pure and applied research, with generosity, and an acute sense for which applications are morally admirable; a discipline of simplicity, eliminating the pursuit of rank or awards or status or recognition — these are still the fundamental components of a culture worth immersing myself in, worth spreading to others. Taken together, they create a deeply rewarding occupation, an occupation that quietly, powerfully, moves us forward, and higher.