by Milton Dawes

Some of my very high order speculating-abstractings at this time-place…

I suggest a lot of visualization and imagination in trying to follow what follows. ‘Rational’, ‘intellectual’, ‘logical’ analysis alone will not help much. It could help a great deal also, to remember that the world does not present itself to us as “variables” and “functions“. These represent “labels” we created for our own convenience – semantic tools to help us communicate with ourselves and others, and understand ourselves-in-our-world. When we forget this, and confuse words with what they represent, we get misunderstandings. And instead of understanding, we get instead of communication confusion.

In Science and Sanity, Alfred Korzybski wrote:

“As the organism works as-a-whole, for its optimum working, and, therefore, for sanity, we need a language, a method, which may be translated into a semantic reaction by which to translate the dynamic into the static, and vice versa; and such a language, such a method is produced and supplied by mathematicians.” (p. 288)

In contemplating this statement, I ask: Could features of the calculus, differentiation of a function, and anti-differentiation, have anything to do with translation from dynamic to static and vice versa? And I answer myself : I believe so. (Remember now that concerning what follows, I have not studied mathematics. And the statements I make represent my generalizations of the method of the calculus – the little I understand of it).

Using speed as an example: “Speed” as movement (dynamic) is translated to the static when we differentiate and get the representation ds/dt. “Speed” (a static label) as a function – a ratio, relating change of position, and change of time. In other words, distance traveled in time taken. This represents one way our limited nervous systems manage changes and relationships, too fast, too complex, too enormous, for us to map directly. We scale things down to more manageable proportions.

If I am running (dynamic), I have an experience I can relate to the term “speed“. The term “speed” could be considered a crude macro-mapping, and verbal representation, an abstraction of the activity I call “running“. So we have muscles, feet, hands, breathing, heart pumping blood, etc., as lower order abstraction-differentiation, to “running” (higher order abstraction-integration), to “speed“, (still higher order). The ratio ds/dt represents a more precise micro-mapping as a mathematical translation, a differentiation, a representation, a method, which can be applied generally to many different situations involving movement. So we have gone from lower order to higher order and back to lower order. And ds/dt (the formula) can be considered a higher order abstraction embodying lower order abstractions (change in position divided by change in time).

Here is another example. Let’s take the term “consciousness“. Again, we have a dynamic situation which many of us can recognize as related to that term. The term represents a macro-mapping, a static verbal representation of an ongoing process; in other words, a continuous function. The activity called “consciousness” (human) can be considered as a system that both differentiates and anti-differentiates. Anti-differentiation can be considered as the reverse of differentiation.As mentioned before and put in very simple terms, our consciousness scale things down and then does what we might call “creative re-assemblings“.

And let’s now take “Whatever Is Going On, in and around us” as another dynamic situation and give it the static label WIGO. We listen, look, select, analyze, objectify (create objects from sense impressions), make distinctions, label, describe objects, make inferences, explain, etc. We could view this as “differentiating-consciousness“. And we also classify, search for meanings, look for patterns, generalize, understand, explain, generate beliefs, create scientific laws, mathematical formulas, come to conclusions, have ‘feelings’ about things, and so on. We could view this as anti-differentiation. It is important to remind ourselves that depending in the level of abstracting, many of these activities can be classified as both differentiation and anti-differentiation. When I listen, I hear-select-abstract a particular sound. But this sound usually represents a complex of frequencies, and so my nervous system has, by its differentiating, integrated all these varying frequencies into what I hear and call “sound“.)

We could now look at consciousness of abstracting as remembering that in our differentiating-abstractings, in everyday situations – as in mathematics – we have left out factors. In differentiating “speed” as the ratio of change in position and change in time, for instance, I left out of the ‘equation’ the clothes I wore, the kind of running shoes, how anxious I was to reach my destination , etc. So in relating this to WIGO, our consciousness differentiates and integrates (anti-differentiates). We ignore-select-abstract factors, and we create-abstract static representations. The “density” (ratio of instances of consciousness of abstracting with respect to time) gives a measure of the accuracy of our mapping, as a function of the unreachable limit of “being aware of all that’s going on“.

The power and the beauty of the calculus for me (the little I know of it) lies in this close set of relationships involving:

variables (a value selected from a given range, abstracting)

functions (relationships between variables)

change

change of change

invariance

rate of change

maxima and minima

changing relationships

time

incremental differences

trends

self-reflexiveness (humans created calculus, and calculus helps us to understand ourselves)

non-identity

non-allness (the concept of approaching a limit necessitating approximation)

mapping

consciousness

consciousness of abstracting

etc. 

I propose that anyone looking at the vocabulary of the calculus cannot fail to recognize how easily these terms can be generalized and meaningfully applied to everyday situations.

If we are not conscious of abstracting, we identify. If we do not discriminate or differentiate, we identify. “We see what we see because we miss all the finer details“, Science and Sanity, p. 376. “Sanity means adjustment, and without the best structural knowledge of each date concerning this world, such adjustment is impossible“, p. 727. “As a structural fact, the world around us is not a ‘plus’ affair, and requires a functional representation“, p. 603. The calculus as a tool works very well in helping us understand ourselves, our world, and ourselves-in-our-world.

The definition of the calculus that I have found very useful is this one: “The study of a continuous function by following its development through indefinitely small steps“. Now consider this … if the continuous function we study is our own behavior with respect to time, one can easily get a feel that the calculus is more than a mathematical device. Since we live in a world of changing relationships, the calculus can also be used as a ‘psycho-logical tool‘. We can apply it to help us study factors related to personal development, improving communication, problem solving, conflict management, time management, stress management, and much more. The method of the calculus can be applied to help us understand and improve almost anything we do. It’s mainly a matter of paying very, very close attention to what we are doing – how we are doing what we are doing – and what happens when we do whatever we happen to be doing.