In his now cult classic book, "Antifragile: Things That Gain from Disorder", iconoclastic thinker Nassim Taleb weighs in on such diverse topics as how the "losers" in history eventually prevail, why Thanksgiving turkeys should not make future predictions based strictly on the past and why an increase in theoretical understanding of medicine actually leads to fewer drugs being patented. The arguments he develops are witty, informed by a wideranging, targeted erudition. I will leave the noninitiated reader to develop a taste for this undefined and indefinable thinker. But the core of his dichotomy pits the idea of "fragility" against that of "antifragility." He then relates "fragility" and "antifragility" to a wide array of states of the world that concern us all, and in particular, the presence of government in our lives.
As budget negotiations in Washington drag on in a state of imperial and splendid confusion, leaving the American people in a state vacillating between mindless stupefaction, voyeuristic fascination and, finally, total and utter disgust, the question arises whether the growth of the debt up to now, independently of what Washington does this year, already exhibits a state of "fragility," or an inability to withstand "black swan" events.
Imagine, for a minute trying to decipher the relation between the growth of the deficit and the debt. The dynamic of the change in deficit would determine the borrowing needs of the government.
Then change in the debt with respect to the deficit would represent the first derivative, which would bear a positive relation. But even more important, to see if we have reached a "tipping point" in the growth of debt, is to examine the change of the change. Is this change of change growing or declining? If the change of the change increases, that represents concavity, if declining, convexity. The latter state, according to Taleb, would represent "antifragility," a more desirable state; if concave, "fragility." (I am broadly interpreting the idea).
If the growth in debt exhibits "fragility" up to this point, the goingson in Washington might turn out to be completely inconsequential  under these dire conditions, the die would already have been cast.
One way, then, to look at whether the rate of debt has reached a "tipping point" is to look at how the concavity itself changes.
Further delving into the matter, I found that, if one were to isolate the study to the last decade, the rate of change of the rate of change stays relatively stable (mostly convex or "antifragile" during the early Bush years), to become concave in 2006, staying concave until 2009, and then gyrating between convexity subsequently. (Between 2008 and 2011, the mean of the measure of convexity/concavity was positive  thus showing net concavity).
Next, I thought it would be interesting to see whether the concavity itself is stable, or whether the rate of change of the convexity changes itself. Taking the mean of this rate of change, I found a value of .0001129 between 2000 to 2011, and a mean of the convexity of .00019599 between 2008 and 2010. Thus, the mean concavity of the relation between the deficit and the debt more or less stays the same over the three years in the Obama Presidency compared to the Bush years  the overall concavity seems to stay stable.
Now examining the same parameters of debt per capita looked at in isolation, (and not in relation to the growth in the deficit), the following table sums up the mean convexities between 2000 and 2010 and the change in the convexities over the same period:
Debt per capita 
Convexity (concavity) 
Mean change in convexity (concavity) 

Between 20002010 
.003 
.000007 

Between 20082010 
.00024 
.000003 
Some good news about the change in debt during Obama's presidency  it seems that the growth of debt per capita, while concave during the whole period between 2000 and 2010 becomes less so between 20082010. But the change in this variable is relatively slight. So, while the tipping point hasn't yet been reached (concavity hasn't been increasing, but not decreasing either), the concavity itself stays constant.
Or while we're still going down the rabbit hole, the hole hasn't visibly contracted, but not expanded either. Or so it seems...
Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.