This is not an article about politics but it is about the Democratic National Convention (D20) held in the last week. Specifically, it’s about one phrase in the speech by Vice Presidential candidate, Kamala Harris (see video). The phrase is “*inflection point*” that was also echoed the following day in the speech by Presidential candidate Joe Biden. It seems that the Democratic Party are hoping that this phrase will be a rallying call for them come Election Day in November.

“We’re at an inflection point. The constant chaos leaves us adrift. The incompetence makes us feel afraid. The callousness makes us feel alone. It’s a lot. And here’s the thing: We can do better and deserve so much more.[1].

I’ve been trying to remember when I’ve heard the phrase “*inflection point*” outside of high school or university mathematics and science. As a former university teacher I certainly used “*inflection point*” in a mathematical context when teaching chemistry. I don’t remember ever using or hearing the term in ordinary conversation. As a single word, *inflection,* can be used to refer to a change in the tone or pitch of someone’s voice when they’re speaking.

The Merriam-Webster Dictionary gives the meaning: “*a moment when *significant change occurs or may occur” or a “turning point.” Another point is that the Qatari broadcaster Aljazeera is reporting these same words as an “inflexion point” [2]. So what’s the difference between an “infexion point” and an “inflection point?” Fortunately, this is an easy question to answer. inflexion is an older British spelling similarly to reflexion or connexion. which are rarely seen in 21st-century writing. The spelling “inflexion” seems to have survived the 20th century in a few places, such as *Aljazeera*. [*As I’m writing this, my spell checker is accepting “inflexion” but rejecting “reflexion” and “connexion”].*

Interestingly, the Merriam-Webster Dictionary [3] gives an example of the usage for “inflection point” from a speech made by Barack Obama at the commencement address for Rutgers State University of New Jersey, 2016 [3]. Evidently, the Democratic Party have history in using the phrase “*inflection point:*” prior to Kamala Harris in 2020.

It depends on us, on the choices we make, particularly at certain

inflection pointsin history; particularly when big changes are happening and everything seems up for grabs. Barack Obama

### A Refresher in Calculus

Do you remember the meaning of an “inflection point” from high school or first-year university calculus? Thought not. So, here’s the refresher. Firstly, there’s a handy piece of free software for calculus, algebra and geometry called *GeoGebra *which makes the maths a lot easier to master.

**Problem: f**ind the inflection point for the function:

**Answer**: Write (or just copy and paste) the function into *GeoGebra*. A chart of the function appears straight away, the red curve. Use *GeoGebra *to find the first and second derivatives for *f(x) * (see the the panel right and the chart below — the 1st derivative chart is switched off because it makes the chart too cluttered).

The 2nd derivative chart is coloured in blue and given by the line *f”(x) = *30*x + *4*.* Where *f”(x)* is negative (*i.e.*, in the bottom half of the plot) then *f(x) *is said to be: “*convex downwards*.” Where *f”(x)* is positive (*i.e.*, in the top half of the chart) then* f(x) *is said to be: “*convex upwards*“. The single point A is where *f(x*) changes from being convex downwards to being convex upwards (where *f”(x)* = 0), this is the *inflection point* (A) ,

The inflection point can be calculated from *GeoGebra* or from where the second derivative crosses 0, f’’(x) = 0, therefore *x *= -4/30 = -0.133.

Though the mathematical meaning is more precise and specific, you can see from the above discussion how the mathematical definition agrees, in general terms, with the point of change meaning that Kamala Harris is seeking to convey. By using the term “*inflection point*” she is indicating a turning point from heading in the direction of *chaos and incompetence*, to a new direction of *certainty and inclusion*.

Interestingly, Kamala didn’t borrow an alternative mathematical term “*step function*” which means a discontinuous jump from one state to a different state. A *step function *might have conveyed much of the same meaning: a jump from *chaos and incompetence* to *certainty and inclusion*. I suspect the term “*inflection point*” was used because it implies a more gradual change than “*step function*” and because it has more resonance with voters.

### Inflection Points in Business

In a time of rapid technological change global business is becoming increasingly driven by the digital environment in which they operate [4]. Strategic decisions around when to adopt new technology products and services are becoming crucial to success and even survival of whole organisations. Especially in this era of COVID. This is where the concept of an *inflection point *comes to play: it’s that point in time when investing in a new technology stands to gain the best return and competitive advantage. The concept is best illustrated with a chart of a *sigmoid* function or S-curve. We’ll use *GeoGebra* to construct the plot, similarly to that for the cubic function above:

The *sigmoid* curve (the green curve) combines aspects of the “step function” but it varies smoothly. The above plot shows the features that are important to a business interpretation of the “inflection point.” Too far to the left in the chart represents the introduction of a novel but immature technology. This is the “*fad*” stage.” When adopting technology at this immature stage, it’s too easy to choose a technology that will be superseded in the short term (the “*wrong horse*“). On the other hand, waiting too long, as shown on the right of the chart, will lead to the adoption of a mature but mainstreamed technology that no longer possesses an edge over the competition.

The middle point of the *sigmoid* curve is where the mathematical inflection point lies. This is the point where the second derivative curve (the orange curve) crosses the *x *-axis. The mathematical inflection point A represents a new technology at its “*next big thing*” stage of development. This is the stage where rapid adoption across the industry sector begins to take place.

The goal of business should be to anticipate wide-spread adoption at the mathematical inflection point A by selecting new technologies at the business inflection point B.

Adopting a new technology at the business infection point B gives the best return on investment by providing a competitive edge.

In summary, we have examined the meaning term “inflection point” from the speech by Kamala Harris at the Democratic Party Convention (D20) from a several perspectives. From the usage of the term in common language, the dictionary definition, its usage in mathematics of calculus and in its most recent usage in business.

After reviewing the term “inflection point” from all of these angles, my understanding is that Kamala’s meaning was similar to the business meaning of trying to anticipate the next big political idea and to carry it forward to Election Day in November. No doubt we’ll be hearing a great deal more of the term “inflection point” over the next month or so. Will we see an “inflection point” election on the first Tuesday in November?

The inflection point may have played well at the Democratic Convention but would they risk it in an electoral campaign where it will inevitably come across as “geek” versus ”freak”?

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*Disclaimer: *I have no connection with the Democratic Party (USA) financially or personally. I have no desire to promote the Democratic Party politically. My personal political views are discussed elsewhere in the article: *My Trip Down Memory Lane*.

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[1] Christina Wilkie, CNBC, “Here’s what Kamala Harris said at the Democratic National Convention,” updated: 20-08-20, accessed: 23-08-20, available online:

https://www.cnbc.com/2020/08/19/dnc-2020-kamala-harris-speech.html

[2] William Roberts, Aljazeera, “Kamala Harris sees US at ‘inflexion point’ in upcoming elections,” published: 20 August, accesses 22 Augsut, available online:

https://www.aljazeera.com/news/2020/08/kamala-harris-sees-inflexion-point-upcoming-200820013321364.html

[3] Barak Obama, The White House, Obama Archives,” Remarks by the President at Commencement Address at Rutgers, the State University of New Jersey,” published: 15-05-2016, accessed: 24-08-2020, available online:

https://obamawhitehouse.archives.gov/the-press-office/2016/05/15/remarks-president-commencement-address-rutgers-state-university-new

[4] Jim Highsmith, Mike Mason and Neal Ford, ThoughtWorks, “Inflection Points,” published: 13-07-16, accessed: 21-08-20, available online:

https://www.thoughtworks.com/insights/blog/inflection-points

### Charts

Links to the mathematics and charts I’ve constructed in *GeoGebra*:

1. https://www.geogebra.org/classic/ej7mebr9

2. https://www.geogebra.org/classic/agczwgvs

Excellent article. This makes the term much more clear than my understanding before when I heard it. Thank you.

Thanks, Marianne. I like the term “inflection point” but if you have to explain it so most people can understand it then somethings wrong.