In Starting Strength, Practical Programming, and The Barbell Prescription we present what I have come to simply call the Rippetoe Plot. I choose this terminology because (a) that’s what it is: a plot, and (b) “plot” sounds so much more congenially nefarious and subversive than “graph.” Notwithstanding any intrigue or derring-do we might impute to it, the Rippetoe Plot is a concise and beautiful representation of the relationships between training, rate of performance increase, and complexity of programming. It is, to quote a great man, self-contained and fairly explanatory:

Note if you will that the performance curve on the Rippetoe plot plateaus as it approaches a horizontal line that Rippetoe has variously called the individual genetic potential (Starting Strength, 3d Ed, pg. 293) or just the potential (Practical Programming, 3d ed, pg. 41). I have sometimes called this boundary the “age-adjusted genetic potential,” but its position is determined by more than just genetic endowment and age. The value of this line is so multifactorial, and so individualized, that I have come to call it the “specific performance potential,” or just the specific potential.

This line is an asymptote: The Athlete’s performance approaches the specific potential, but never acquires it.  

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So what, exactly, makes an asymptote? The concept comes to us from mathematics. If we graph a function, and find that the graph comes arbitrarily close to a line in the plane, then that second line which it approaches is an asymptote.

Rational functions are great examples. A rational function is an expression in the form of a fraction, in which both the numerator and the denominator are themselves algebraic expressions (technically polynomials). For example:  

Is a rational function. Asymptotes are characteristic of rational functions. A couple of examples will make things clear.

Consider one of the simplest of all rational functions: y = 1/x. It is a trivial matter to plot values for this function. Set x equal to any number, and divide one by that number to get the corresponding value for y. Make a table:  

x	y
-4	-1/4
-2	-1/2
-1	1
0	UD
1	1
2	1/2
3	1/3
4	1/4

This table clearly shows trends in the function. The function takes a value y for all x, except at x = 0. At x = 0 we have y = 1/0 which, contrary to common misconception and error, does not equal either zero or infinity. At x = 0, y is undefined. At any number greater than or less than zero, y does have a value – although as x approaches zero the value of y becomes arbitrarily huge. As x approaches zero the value of the function approaches infinity – but it never gets there. So we say that x = 0 is an asymptote of the function.

We also note that, as x increases, the value of y becomes arbitrarily small. We say that y approaches zero as x approaches infinity – but it never gets there. So the line y = 0 (which is the x-axis) is also an asymptote for this function.

In other words, there are no values for x that give y = 0.

Thus, the equation y = 1/x has both vertical and horizontal asymptotes – y is undefined at x = 0, and x has no value that yields y = 0.

This is all easy to see when we graph the function:  

Now you can see, if you didn’t already, what I mean when I say that the Rippetoe plot displays an asymptote. A trained performance attribute approaches the specific potential asymptotically. We may – with good training, good nutrition, good recovery, and above all good luck – approach our genetic potential. Indeed, exceptionally dedicated and exceptionally fortunate individuals may come arbitrarily close to their potential. Along the way, we will learn a great deal and rack up tremendous accomplishments. And our explorations, in the nominal adaptive zone and beyond, will raise many questions. Not the least of these questions is: How may we best train to exploit our gains as we approach the asymptote of our performance?

Most importantly, we will find that our approach to the horizons of performance will reveal us. It is when we approach the asymptote that our training and performance become the most individuated and unique. Our approach to the the specific potential elevates us and illuminates who we are. But we shall never achieve it.

Thus, the Rippetoe plot is, in this respect, like the graph of a rational function.

This concept of an unassailable horizon – so clear in the asymptotes of rational functions – finds a compelling parallel when we come to analyzing problems of a different order. Algebraic functions aren’t the only things we can rationally analyze. In modern thought, the theory of knowledge known as rationalism was highly associated with mathematics, as in the philosophy of Descartes, Leibniz, and Spinoza. On this view, we can learn about the world through reason: logic, mathematics, analysis.

We can rationally approach any of a number of Big Questions: What is the ultimate nature of reality? What is the meaning of life? What state, if any, preceded the observable universe? Why is consciousness like this? What happens to our consciousness after death? Does God exist? If He does, what is His Nature? What is the nature of the Good, and of Justice? What is beauty? Why does anything exist at all?

All these questions have been interrogated, most fruitfully I think. From exploring these issues to the limits of human reason, we have learned much, not least about our own finitude. But notwithstanding the generative power of such questions, none are resolved.  

Indeed, none of them even promise to be resolved. Like rational functions approaching the limit at which the denominator goes to zero, such asymptotic questions become undefined at the limits of interrogation. Like the graph of y = 1/x, a question like, What is the good? or Why does anything exist? always approaches a point where it takes off to infinity, presenting a barrier impenetrable to rational inquiry.

Kant puts this problem beautifully in his First Critique, the Critique of Pure Reason, at the very beginning, when he says,

“Human reason, in one sphere of its cognition, is called upon to consider questions, which it cannot decline, as they are presented by its own nature, but which it cannot answer, as they transcend every faculty of the mind.”

(Preface to the First Edition, 1781, as translated by Meiklejohn)

Doubtless some of the more doctrinal among us may object that a particular tradition, religion, culture, or philosophy absolutely does know the answer to one or more of these questions. Indeed, some religions, in the arrogance characteristic of that most human of institutions, claim to answer all these questions.

But such claims always prove hollow, completely refractory to either rational or empirical demonstration. At the limits of logic they break down into syllogisms that are either invalid or unsound, at the limits of analysis they yield the contradictions that Kant called antinomies, at the limits of scientific investigation they are found to be beyond adjudication by any critical experiment, and in practice they are far from categorical or universal.

These are the Big Questions, the questions that haunted us long before the Greeks gave them their classic form, and which haunt us still, precisely because they resist disposition by reason, evidence, or even general acclamation. Religion, science, and philosophy give us manifold, beautiful, fecund, speculative answers to the Big Questions. But never The Answer.

These speculative answers are of course far more speculative than they are answers. They are sublime graffiti on the asymptotic walls that surround us in every direction, in every dimension of our existence. When we claim that our tradition or our philosophy or our culture has the answer to such a question, we’re simply adding to or endorsing some of that graffiti.

A great thing about that: we can’t really be wrong to do so, unless we contend that this Scrawl-On-The-Wall is The Answer, more legitimate than any other.

Stamp your feet and howl if you want, but if we had an incontrovertible metaphysical source or standard for the Good, or the Beautiful, or for Being, we would know it by now. And Kant himself did not find that the “tragedy of reason,” as it has been called, was any barrier to faith or the elaboration of a deontology of ethics (“Practical Reason,” as opposed to “Pure Reason”). But he also made it clear that it was precisely the limits of reason that made room for The Important Stuff that lies beyond reason. Two hundred years later, Wittgenstein would say essentially the same thing about logic and the mystical (Tractatus Logico-Philosophicus). But we don’t need dead German philosophers to tell us what we all know: The human arena is walled by mysteries.

The asymptotes of reason, logic, and science don’t extinguish or exile The Important Stuff – they mark and illuminate the magnitude of The Important Stuff.

Even so does the specific potential on the Rippetoe Plot delimit and illuminate what is important. As in the asymptote of an algebraic function, the limit of pure reason, or the horizon of logic, it tells us where we can expect to hit the proverbial wall. But like all the other asymptotes that define our lives, it doesn’t tell us exactly where or when or how each of us, as an individual, will hit that wall. And it doesn’t tell us what we can or cannot write on that wall.

It just shows us where the questions get big: How are we to train when we encounter it, at the far end of the nominal adaptive zone and beyond? Rippetoe’s Graph can’t tell us, because it is at those limits that our training is most completely our own.

Like the empty asymptotic walls for the other Big Questions that are just begging for you to scribble and paint and project your answers upon them, the specific potential asymptote calls for your answer, any answer that keeps you close to the wall, and perhaps ascending it, even though you can never scale it or go through it.

Here Be Dragons. Here your training goals, your programming, your practice, your metrics for meaningful progress are more completely and uniquely yours, at the threshold of the impenetrable, than at any other time in your training career. Here is where you make the choices that define what your training means. There is no logic, no science, no program, no book, and no coach that can choose for you. It’s entirely up to you.

Here at your specific potential, as at the other asymptotes of your existence, you the Athlete become most truly and uniquely yourself, where you are most constrained but also the most liberated, where you may, in a rather less melancholy way than Hamlet, proclaim that though you be bounded in a nutshell, you may yet be a king of infinite space.  

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