The dialectic of extremes and means is a method of reasoning whereby one begins with extremes and reasons to means or *vice versa*. If one begins with means, these are considered as unanalyzed entities, attributes, propositions, etc. The goal is to work out the implications of them as principles or to analyze them into their constituent parts as a combination of extremes. If one begins with extremes, these are considered as unsynthesized entities, attributes, propositions, etc. The goal is to synthesize them into their fullness and completion as integrated means or to work from partial truths toward full truths.

We live among means, that is, we live in the middle ground, a mesosphere where things are muddled and messy but familiar and common. Philosophy is often said to begin here, with what is commonly known rather than with specialized knowledge. Whatever we find must come back to the middle ground where we live or else it is like a dream unrelated to our lives.

Classical deductive logic works from truths to their implications while preserving truth. It assumes that truth is known at the beginning, that truths are known in the middle ground of life. They may be known because they are axiomatic (worthy of assent) or because they are self-evident, or because they were given by a trustworthy source. The outworking of such truths leads toward extremes.

The dialectic of reasoning from extremes to means is focused on the end, not the beginning. It does not follow from truths; it leads toward truths. One does not usually begin with truth. One usually begins with something at hand, something muddled and messy. Truth is something that must be sought. This dialectic begins with partial truths and reasons toward full truth.

Extremes express simple but partial truths. Proverbial statements often express extremes – that’s why there are often contrary proverbs. For example, the Book of Proverbs includes these two: Do not answer a fool according to his folly, or you will be like him yourself. Answer a fool according to his folly, or he will be wise in his own eyes. (Pr. 26.4-5)

There are many pairs of entities, attributes, propositions, etc., which express contrary extremes and are partially true. For example, a preference for simplicity leads to extremes: in classification and typology, the extremes are all elements in one class and every element in its own class. Some people (called lumpers) tend to combine elements into fewer classes and others (called splitters) tend to split elements into more classes. Who is right? They are both partially right.

Reasoning from extremes to means may be deductive or inductive. The deductive form works via a form of backward chaining. It starts with a mean which is a hypothesis or goal and works backwards from the consequent to the antecedent to see if the extremes will support this or any of these consequents. Instead of reasoning from truth, it is reasoning from partial truths. The result is a combination of partial truths, which together form a complete truth.

As an illustration of reasoning from extremes to means, consider arithmetic. Start by defining numbers recursively: if x is a number, then f(x) is a number. For example, if x is a number then x+1 is a number. (Addition could be left undefined at this point but let’s assume it is ordinary addition.) Next, consider the extremes: what are the first last numbers, if they exist?

Answer A: There is a first number; call it 0. There is a last number; call it 2, where 2 is 0+1+1. This is arithmetic modulo 3.

Answer B: There is a first number. Call it 0 (or 1, if you prefer). There is no last number in the sense that there is no unique last number (the sequence must not converge and one can stop at any number arbitrarily). We conclude that 0+1 is a number, as are 0+1+1, and so on in sequence without end.

Answer C: There is no first number in the sense that there is no unique first number. There is a last number which depends on the recursion and the arbitrary first number (called the seed number). The sequence must be convergent. For example, let the seed number be 1 and the recursion such that if x is a number, then the reciprocal of x+1 is also a number. This leads to the sequence 1, ½, 2/3, 3/5, 5/8, and so on. The last number of this sequence is (-1+√5)/2, sometimes called φ (or 1/φ). Notice that other seed numbers could lead to the same last number.

In these examples the numbers formed by the recursions are the means. The extremes (those directly stipulated as numbers or used as seed numbers or formed by sequences) are not really numbers. From ancient times a number has been defined as a multitude so the first number is the second member of the number sequence and there is no last number. The extreme numbers are the limits of ordinary numbers. Ordinary numbers are analogous to partial truths, and extreme numbers are analogous to full truths.

These examples lead to the observation that sometimes the extremes may be contrary in different and multiple ways. First and last are natural extremes but other attributes may be contrary, too: definite and indefinite, arbitrary and determinate, convergent and divergent, etc.

Conjecture: convergent and divergent sequences may be put into one-to-one correspondence.

November 2013