I want to outline briefly certain standard mathematical operations and their opposites where such exist, and suggest an addition to the list. I am aware that this listing does not exhaust the field of mathematics and I do not intend to do so. For example, I say nothing about series, permutations and combinations, probability theory, networks, groups, or any system of geometry or topology.
The potency dimension (strong-weak) is also relevant to survival. If I meet a danger, I need to know its potential strength in order to do damage control. A weak danger, like a mosquito, I can choose to ignore if I have more important things to do, but I cannot shrug off an encounter with a sabre-tooth tiger.
The same can be said about “active-passive” or “fast-slow”. An earthquake may be a stronger danger than a tiger, but it is not descending on me right this minute. (Unfortunately, humans have a tendency to postpone reaction to “slow big dangers” like climate change, which may prove to be our undoing. )
A theological argument about the nature of the Devil is relevant here. A Manichean Devil is “bad, strong, and active”, while an Augustinian Devil is “bad, strong, but passive”. What this seems to be saying is that the Devil does not actively scheme to destroy us (though in many popular legends, like Faust and Don Giovanni, he does), but allows inaction to destroy us automatically. As Burke said, “All that needs to happen to make evil prevail is for good men to do nothing.” This is related to the strong but passive role of Entropy as an Anti-Life Force. In social change movements, we like to say that we are not so much fighting against reaction as against inaction.
Among the dualistic religions, the Zoroastrians probably believe in a Manichean Devil and the Christians in an Augustinian Devil. After all, the Augustinians “won” the ideological dispute with the Manicheans, who were declared to be heretics. The Devil in Christianity, though usually (at least in popular myth as opposed to official theological doctrine) not considered to be entirely inactive, is at least less active than God. The Devil is also seen to be as not as strong as God, who after all is omnipotent. The Devil and his cohorts, e.g. in exorcism rites, retreat when faced with the sign of the cross or the sprinkling of holy water, which are only symbols of God’s omnipotence, not even His actual presence. Satan or Lucifer was only an archangel before his revolt, not the all-mighty ruler of Heaven. Zoroastrian dualism, as I understand it, is more symmetrical — it is almost a bi-theistic rather than a mono-theistic faith, a bipolar relationship of two superpowers.
However, even in Christianity, the Manichean streak still persists; as when, during the holocaust of the witches, the ecclesiastical authorities and the common people spoke darkly about “the principalities and powers” when referring to the Devil’s machinations in seducing women to serve him. The Devil was in fact seen as heading up and controlling a rival “Evil Empire”. Reagan’s calling the Soviet Union an “Evil Empire” was a prime example of the demonization of the enemy which almost enmeshed all humanity in a nuclear war.
What about the role in the linguistic universe of other parts of speech — conjunctions and prepositions? Their role is minor; they serve only as linkages between the main linguistic actors. However, by indicating precise relationships (above or below, in or out, before or after, with, without, or against), they are indispensable as clarifiers of meanings. My separate essay entitled “If-then” discusses the logical, causal, inferential, and moral meanings of what in logic is called the “implication”, a whole series of categories of meanings that would be impossible to conceive without the help of the two little words, “if’ and “then”.
Finally, the simple exclamations “oh”, “wow”, “psst” and the like, are probably the first speech sounds ever uttered, the way the whole linguistic universe was firstborn at the human dawn.
1. Addition and its opposite, subtraction. Since addition is commutative, there is only one opposite. To make subtraction always possible, zero and negative numbers had to be invented. Zero is a kind of pivot in these two operations: the only element whose addition or subtraction does not change the other number. But Boolean addition or its use in symbolic logic has properties different from ordinaIy arithmetical addition. Specifically, it means union: both A and B and the overlap of A and B.
2. Multiplication and its opposite, division. Since multiplication is commutative, there is only one opposite. Multiplication is derived from repeated addition, but it has also other meanings, e.g. in geometry going from a line to an area or on to a volume or higher dimensions. To make division always possible, fractions had to be invented. The number I is a kind of pivot in these two operations: the only element whose use in multiplication or division does not change the other number. Boolean multiplication, or its use in symbolic logic, has properties different from ordinaIy arithmetical multiplication. Specifically, it means intersection: only the overlap of A and B.
3. Raising to a power and its opposite, taking roots. Since raising to a power is not commutative, (i.e. aX$. xa), there are two opposites: taking roots and taking logarithms. But we save the latter for the next section, because it also has another opposite. To make the taking of roots of positive numbers always possible, we have to invent irrational numbers. To make the taking of roots of negative numbers at all possible, we have to invent imaginary numbers. There is no kind of pivot and no Boolean or symbolic-logic analogue.
4. Exponentials and logarithms. While “raising to a power” means x8 , by “exponential” I mean aX, where a is a constant and x is a variable. Taking logarithms is the opposite of exponentiation, i.e. x is the logarithm of aX to the base a. In practice, “the base of logarithm is usually either 10 for decimal logarithms or e (2.732…) for “natural” logarithms. While the first 30perations give rise to algebraic functions (polynomials), this and the next operation give rise to transcendental functions.
5. Trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) and their opposites, arcsin, arccos, etc. These are also transcendental functions. While independently defined geometrially with respect to the circle, they have a deep relationship with exponentials through infinite series, De Moivre’s theorem, and oscillations and asymptotic approaches to equilibrium as solutions of certain first-degree differential equations. (See essay “The Key” in Section x.)
6. Differentiation and its opposite, integration. (Some would reverse this order, because integration seems more positive; but differentiation is simpler and is usually taught first.) These two opposites are profoundly complementary concepts in biology (embryonic development) and political theory (federalism and subsidiarity). They imply continuity and limits. The discontinuous analogues are the “difference calculus” and the summation of series or sequences.