Big Questions, Asymptotes, and the Rippetoe Plot


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.  


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:  

rational function graphed

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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