On another website, I've posted a MATHEMATICS DICTIONARY whose FRONTPAGE carries another paraphrase of the Nursery Rhyme about "... a litle girl Who had a curl Right in the middle of her forhead. And when she was good ...." But there are two significant differences between the one on that Frontpage and the one on the present Frontpage. In the math case, the subject is in quotes, since I'm talking about what too often passes as mathematics, not what math properly is. In the present case, I omit quotes from the subject, since I'm dealing with what I regard as the essential nature of this field. In the other case, I leave "bad" without quotes, since I regard abuses of mathematical usage to be actually bad. In the present case, I put the word in quotes since I really mean the limitations of logic and intend to show how these limitations can be bypassed.

I'll explain this by reviewing the notion (elsewhere) that "physics is corrigible and mathematics is incorrigible", which I owe to the Australian philosoper, Douglas Gasking. To say, "physics is corrigible" means that, when an statement or prediction of physics is shown to be incorrect in application, then physics is corrected. (Its history is replete with such instances. In the past century, Relativity Theory and Quantum Theory represent major and rewarding corrections.) On the other hand, to say, "mathematics is incorrigible", means that when a mathematical model fails in application, we do not change mathematics, but seek another model.

I give an example to show how rewarding this differing policy can be. Arithmetic says, "2 + 2 = 4". We find application after application wherein this model is correct: coins, cars, persons, most units of measure, etc. But I show a case wherein the model fails. If you take two cups of water combined with two cups of water, you find that this exactly fills a four-cup measure. Similarly, with oil. But when you combine two cups of water with two cups of oil, you find an observable departure from the four cup measure. Why? The molecules of water are different in size from the molecules of oil, so do not combine in the same way. You obtain a similar result when you combine two barrels of tennis balls with two barrels of ball bearings. Some of the bearings slip between the interstices of neighboring tennis balls wihout extending the total occupancy.

What happened? The failure of this must have led humans to the concept of molecule, diverse within the diverse chemical elements. (Elsewhere, I've described this as "The WIN-WIN FIGURE&GROUND STRATEGY": You "hold up" a hypothesis or model as FIGURE against the GROUND of REALITY. If it fits, you WIN; if it doesn't fit, it induces a question you would not, otherwise, think of asking, perhaps leading to an answer that advances knowledge -- also a WIN.

Is Logic corrigible or incorrigible? We must regard Logic as incorrigible, as with mathematics. But another STRATEGY enables us to deal with the limitations of logic (described in another file). It's what the Canadian mathematician, Z. A. Melzak, taught us: BYPASS. Z. A. Melzak, wrote a book, Bypassing, A Simple Approach to Complexity, about the principle of conjugacy in mathematical group theory -- which Melzak now applied in various daily, technical and scholastic ways.

The bypass strategy bypasses a difficult or "impossible" problem or situation by accomplishing the same intent through solution of another problem. Melzak shows how BYPASSING can be used in math, science, technology, and in daily life problems. It is perhaps our most useful strategy and Melzak speculates that hominid became human by internalizing bypassing. It readily lends itself to graphing:

  Difficult or Impossible Problem
     -------------------->
turn-|                   ^translate
ing  |                   |answer or
it a-|                   |solution into
round|                   |original problem
     V-------------------^
      solve new problem 

A familiar example occurs in the 1963 film, The Great Escape, the problem of escaping across the compound under guntowers was bypassed by (1) digging down from a dormitory, (2) tunneling horizontally under the compound, (3) tunneling up outside to escape into the woods:

         Escaping across compound
          -------------------->
 tunneling|                   ^tunneling
 from dorm|                   |up outide
down below|                   |to escape
  compound|                   |into woods
          V-------------------^
          tunneling undergound  

Similarly, we can BYPASS the limitations of LOGIC (discussed in another file) by transforming to a process of reasoning unfortunately labeled "fallacy of asserting the consequent" (FAC), which gives us a means of arriving a judgments impossible within Logic itself:

          Reasoning impossible in Logic
             -------------------->
 transforming|                   ^calculate
 to reasoning|                   |measure of
       by FAC|                   |reaoning to
             |                   |advise decision=making
             V-------------------^
          formulate transformed problem  

Table of Bypasses.

Now, we study a file detailing the limitations of Logic. Then study a file showing how to BYPASS these limitations.