"Curiosity, a healthy curiosity, and a desire to learn is what leads the scientist from one problem to another. Once that feeling is lost and there is no excitement or pleasure in learning new facts, the scientist will no longer be able to discover anything new." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Demonstrative reasoning differs from plausible reasoning just as the fact differs from the supposition, just as actual existence differs from the possibility of existence. Demonstrative reasoning is reliable, incontrovertible and final. Plausible reasoning is conditional, arguable and oft-times risky.
"Every meaningful mathematical theory is a reflection of reality: the mathematician idealizes concrete phenomena into a rough scheme when he creates the starting propositions of a theory; later, when he already has drawn certain conclusions, he compares them with the phenomena of reality." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Every science is permeated with proofs, and to the same extent as mathematics, for demonstrative reasoning is an integral part of mathematics." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Extrema is the generic term for the concepts 'maximum' and 'minimum' , like 'parents' is the generic term for 'father' and 'mother'. Extremal problems have to do with finding maxima and minima. We encounter them everywhere. It is hardly an exaggeration to say that all problems solved by living organisms are those involving a search for extrema.
"Generally speaking, when a mathematician introduces a new
term, he ordinarily pays little attention to whether it has a contrasting term
to go with it. For instance, there is a class of ordinary differential
equations but there are no 'extraordinary' differential equations. Actually,
ordinary differential equations are equations in one independent variable,
whereas differential equations involving many independent variables are termed partial
differential equations and not extraordinary differential equations. [...]
"In mathematics the problem of the essence of proof has been thoroughly worked out and every mathematician must master the methods of demonstrative reasoning. Appropriate rules have been established for this purpose. These rules and the concepts of rigour and exactitude of reasoning vary from century to century, and at the present time every mathematician knows the level of rigour of modern mathematics." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"In most engineering problems, particularly when solving
optimization problems, one must have the opportunity of comparing different
variants quantitatively. It is therefore important to be able to state a
clear-cut quantitative criterion."
"In plausible reasoning, one must distinguish a reasonable
conjecture from a less reasonable conjecture and be able to substantiate the
conjecture with the available facts, to find these facts, to search
painstakingly for facts that contradict the conjecture, and to correlate the
findings and again return to plausible arguments.
"Induction is aimed at revealing regularities and
relationships that are hidden behind the outer aspects of the phenomena under
study. Its most common tools are generalization, specialization, and analogy.
Generalization arises from an attempt to grasp the significance of observed
facts and is then verified by further particular cases.
"Induction is the process of eliciting general laws via observation and the correlation of particular instances. All sciences, including mathematics, make use of the induction method. Now, mathematical induction is applied only by mathematicians in the proof of theorems of a particular kind." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Mathematics is the sole avenue for learning how to reason via proof. On the other hand, one must also learn how to conjecture.[…] In a rigorous case of demonstrative reasoning, the main thing is to be able to distinguish proof from conjecture, justified proof from an unjustified attempt." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Mathematical knowledge is fixed securely by means of demonstrative reasoning, but the approaches to such knowledge are strewn with plausible modes of reasoning." (Yakov Khurgin, "Did You Say Mathematics?", 1974)
"Mathematical statistics does not only study procedures for analysing experimental findings but also elaborates methods for taking decisions under conditions of uncertainty, the uncertainty being such as is characterized by statistical stability."
"Maximum and minimum always exist together: if our cup-like surface is turned over, we get a cap, in which the highest point (maximum) corresponds to the lowest point of the cup (minimum). By climbing to the uppermost peak of a mountain we can find ourselves (via reflection in a nearby lake) in the lowest point of the valley below. Here, the mathematician calmly reasons to within an accuracy that amounts to the opposite, so to say, for if we find a maximum and then view the situation from another angle, we see a minimum. The answer thus depends solely on how we view the surface. That is why we always speak of seeking an extremum and not, separately, a maximum or a minimum.
"Most people imagine mathematics to be a deductive science in which all theorems, results and facts are obtained via logical reasoning by proceeding from certain starting axioms, primal assertions, assumed to be self-evident or not requiring any proof.
"[…] under plane transformations, like those encountered in
the arbitrary stretching of a rubber sheet, certain properties of the figures
involved are preserved. The mathematician has a name for them. They are called
continuous transformations. This means that very close lying points pass into
close lying points and a line is translated into a line under these
transformations. Quite obviously, then, two intersecting lines will continue to
intersect under a continuous transformation, and nonintersecting lines will not
intersect; also, a figure with a hole cannot translate into a figure without a
hole or into one with two holes, for that would require some kind of tearing or
gluing - a disruption of the continuity.
"The model of an object, process or phenomenon is some other
object, process or phenomenon having certain features in common with the
original. It is ordinarily assumed that the model is a simplified version of
the object of study. However, it is not always easy to give precise meaning to
the concept 'simpler than the original', for the simple reason that
in reality all entities or phenomena are infinitely complicated and their study
can be carried out with differing and constantly increasing degrees of accuracy."
"The point is that every experiment involves an error, the
magnitude of which is not known beforehand and it varies from one experiment to
another. For this reason, no matter what finite number of experiments have been
carried out, the arithmetic mean of the values obtained will contain an error.
Of course, if the experiments are conducted under identical conditions and the
errors are random errors, then the error of the mean will diminish as the
number of experiments is increased, but it cannot be reduced to zero for a
finite number of experiments. […] The choice of entities for an experiment must
be perfectly random, so that even an apparently inessential cause could not
lead to erroneous conclusions."
"Today, science in a number of cases is able to indicate the
rules or, as the accepted term is, the strategy for making the best (or a
sufficiently good) decision. Under other circumstances, there is no such
strategy, but there are certain recommendations on how to pose questions in a
more reasonable fashion, how to construct a suitable mathematical model of the
situation and how to study the model."
"Thus, the construction of a mathematical model consisting of
certain basic equations of a process is not yet sufficient for effecting
optimal control. The mathematical model must also provide for the effects of
random factors, the ability to react to unforeseen variations and ensure good
control despite errors and inaccuracies."
"[…] topology, a science that studies the properties of
geometric figures that do not change under continuous transformations.
"[…] when a mathematician demands rigorously logical proof
about any assertion, he does so not for his own pleasure but to verify the facts,
which might easily appear to us to be obvious but which, when verified, prove
to lie erroneous.
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