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prev "Autoformalisation - Knowledge acquisition of professional skills" by Gregory Gromov,
   Microprocessor Devices & Systems, Moscow, 1986, N 3, p.80--91, Chapter 6
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Knowledge Autoformalization and the Criteria of ‘True Science’

Speaking before various academic audiences we have frequently had to answer one and the same question: “Why did you need to introduce this new term, ‘autoformalization’, in order to single out a certain set of creative elements used in computer work? If nothing else, it is a logical paradox! Regardless of the tools available, how can a person give a formal description of something that he is unable to explain even to himself? And you truly must admit that your reasoning consists of a lot of elements that are outside the boundaries of true science".

The wording of this question differs, but the main thrust remains amazingly constant. Small wonder that as soon as my book (G. R. Gromov, National Information Resources: Problems of Industrial Exploitation. — M. Science. 1984. 240 p.) was published, the chapter “Professional Knowledge Autoformalization Techniques“ provoked criticism, mainly from these same, purist positions. Naturally, it is too difficult to give short answers to all questions that arise in the course of such discussions. That is why we shall only try, without dwelling on the emotional aspects of the so-called ‘scientific character’, to give priority to aspects of the knot of methodological contradictions that reveal themselves in this context.

Naturally, neither PCs nor any other tools can themselves drastically change the borders of professional knowledge ‘zones of certainty’, formed during the long history of development. If today we speak of the possibility of autoformalizing professional knowledge using PCs, it must be clear that, in this case, the very notion of ‘formalization’ undergoes significant transformation. Evidently, this should be repeatedly emphasized in order to avoid the terminological misunderstandings that tend to arise in this context.

On the other hand, any attempts to approach the perimeter of the ‘golden temple’ of exact sciences—these monopolies on methods used by traditional mathematics in professional knowledge formalization—have until now been regarded by the academically decorated guardian priests of its outposts as heresy bordering on sacrilege.

 

...the application of mathematical methods in medicine, though it has a relatively long history, is still in its initial stage.

When we started working with real medical data, it became clear that the reliable general principles that mathematicians had applied in solving physical and technical problems were of little use in this new field. The situation is evidently very much the same in other fields where mathematical methods are not traditionally used.

~ Israel Gelfand

 

Regardless of the specific subject or form of such discussions, their conceptual line of reasoning remained, as a rule, exactly the same: "Is this mathematics? Is this even science?“

Clearly, we cannot dwell here in any length on the various examples of the stability of the myth about unshakeable criteria of ‘strict formalization’ and their being synonymous with the very notion of a scientific method, which is of considerable interest to the history of science. The reader can find descriptions of bitter scientific clashes, brilliant ascents, tragic deadlocks, and practical utilization in the monograph by Soviet mathematicians Ilya Blekhman, Anatoly Myshkis, and Y.G. Ponovko, "Mechanics and Applied Mathematics: the Logic and Peculiarities of Mathematical Applications“. The American mathematician Morris Kline’s book, Mathematics. The Loss of Certainty, which has recently been translated into Russian, contains an analysis of the situation from the point of view of the historical development of both mathematics and the natural sciences, as well as extensive factual material allowing the reader to draw his own independent conclusions. As the author noted in the introduction, “Our predecessors regarded mathematics as an unsurpassed model of strict argument and a collection of elements of knowledge presenting ‘the ultimate truth’ and true knowledge of the laws of nature. The main topic of this book is a story of how Man came to realize these notions to be erroneous and developed a modern understanding of the nature and role of mathematics".

Now we shall briefly illustrate the general meaning of autoformalization using a very simple example. Let us suppose that we have installed a computer on a cross-country vehicle that travels on caterpillar treads. We have given the vehicle environment-sensing transducers and developed the necessary hardware and software to guide this self-propelled unit.

Statement of the problem: there is a hut in the middle of a treacherous swamp. A local person stays in this hut periodically, while hunting, and returns to the hut each time without particular difficulty, as he knows how to choose a relatively safe path. He is a professional driver and, if called upon, could drive our vehicle to the hut.

Question 1: If a team of qualified mathematicians and programmers studied the control algorithm for this vehicle and watched the driver taking the vehicle to the hut and back several times, and were provided with open communication with the driver, would they be able to write a program so that the vehicle-borne computer could automatically guide the cross-country vehicle along the same route?

 Judging by the reaction of the audience, most of the people present here have no doubts of the answer to the question: This is impossible! It is clearly not the peculiarities of a certain part of the terrain that pose the main obstacle. Notwithstanding the great qualitative advance in navigational equipment available on vehicles today, including the latest in onboard computers, we have to admit that the pilot’s profession is still far from becoming obsolete. Moreover, the above-mentioned difficulties are not unique to such transport problems.

“It is quite natural," Jacques Hadamard noted in this regard, “to speak about a more intuitive type of mind for which the area of idea combinations lies more deeply, and a logical type of mind, when this area is situated closer to the surface."

Even casual, friendly discussions with the ‘pilot’—the leading  industrial engineer or any other top level specialist in a structurally undeveloped field of knowledge—can contribute very little to helping the programmer understand the subject matter of the problem. Actually, the time frame needed for a more or less thorough understanding of non-trivial problems of this sort is a lifetime. That is why in practice we usually have an alternative: either to label a difficult problem ‘not yet ready for automation!’ or follow the poet's example: “Here is my pen—you may write poetry yourselves". Let us try to follow the probable development of events in the latter case.

Suppose that the team of programmers involved in solving the cross-country vehicle problem stops trying to work out an algorithm for analyzing the imperceptible ways of assessing the situation in the swamp. Instead, they would try to write the required basic software—to install drivers to control the cross-country vehicle’s navigation system, to provide communication with environmental transducers, etc., using a popular high-level programming language, and then invite the hunter to sit at the vehicle’s onboard computer console in order to try and write a program of piloting the vehicle to reach the hut, implementing his or her own (still unconscious) navigational algorithm.

Let us also assume that the hunter has already learnt the fundamentals of this ‘second literacy’ and is able to start working at the computer console using BASIC, for instance. The hunter will spend several days or weeks getting acquainted with the specific version of the language and with the built-in, application-oriented functional extensions. Then the hunter will be able to start writing a program for piloting the vehicle in the initial part of the route, which is the least difficult. A long and exhausting debugging procedure will follow. We can imagine roughly how this might be done.

"Go!“ The vehicle advances several meters and sinks into the bog. We then lift the vehicle and tow it to the starting point, analyzing the situation. The author of the program carefully examines the ground, following the caterpillar treads’ tracks up to the spot where the tracks disappear under the water, and then sits for a long time, reading through the program line by line. “Aha, I think I’ve got it.  The information in the program was incomplete: choosing the direction with orientation to drier moss is only acceptable within areas of water formed from melted snow. But shortly after the rain it is also necessary to take into account the color of ice-free water patches and direct the vehicle to more ’reddish’ spots, where the ground usually happens to be more firm". Corrections are made in the text of the program; preliminary testing is conducted using a computer simulator ‘text map’ of the terrain. Again the engine is warmed up, and the command is heard—“Forward!” This time the vehicle goes a little further before dropping into the muck, and so on.


 

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 "Autoformalisation - Knowledge acquisition of professional skills" by Gregory Gromov,
   Microprocessor Devices & Systems, Moscow, 1986, N 3, p.80--91, Chapter 6


  Copyright © 1986-2011 Gregory Gromov