Discrete Encounters

Craig Bauer

• Éditeur: Chapman and Hall/CRC
• Année: 2020
• ISBN:
  9780429682889
• Édition: 1
• Sujet: Mathematics
• Sous-sujet: Counting & Numeration

Eschewing the often standard dry and static writing style of traditional textbooks, Discrete Encounters provides a refreshing approach to discrete mathematics. The author blends traditional course topics and applications with historical context, pop culture references, and open problems. This book focuses on the historical development of the subject and provides fascinating details of the people behind the mathematics, along with their motivations, deepening readers' appreciation of mathematics.

This unique book covers many of the same topics found in traditional textbooks, but does so in an alternative, entertaining style that better captures readers' attention. In addition to standard discrete mathematics material, the author shows the interplay between the discrete and the continuous and includes high-interest topics such as fractals, chaos theory, cellular automata, money-saving financial mathematics, and much more. Not only will readers gain a greater understanding of mathematics and its culture, they will also be encouraged to further explore the subject. Long lists of references at the end of each chapter make this easy.

Highlights:

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• Features fascinating historical context to motivate readers

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• Text includes numerous pop culture references throughout to provide a more engaging reading experience

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• Its unique topic structure presents a fresh approach

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• The text's narrative style is that of a popular book, not a dry textbook

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• Includes the work of many living mathematicians

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• Its multidisciplinary approach makes it ideal for liberal arts mathematics classes, leisure reading, or as a reference for professors looking to supplement traditional courses

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• Contains many open problems

Profusely illustrated

1

Logic

The fact that all Mathematics is Symbolic Logic is one of the greatest discoveries of our age; and when this fact has been established, the remainder of the principles of mathematics consists in the analysis of Symbolic Logic itself.

Bertrand Russell, Principles of Mathematics, 1903¹

If you are engaged in an argument and forced to admit that your opponent’s viewpoint is logical, you are in trouble. The only thing worse would be if he or she presents a proof. In that case, no response is possible. Mathematics is the only discipline that can claim proof. Therefore, it is important to develop a deeper understanding of what that means. The first step is to understand logic.

Credit for first establishing logic as a field of study goes to Aristotle (Figure 1.1). Socrates and Plato are sometimes cited, but they did not look at things as abstractly (generally) as Aristotle. Appeals may even be made going back farther to the Pythagoreans, for example, or others. This is not appropriate. To understand why we must make a subtle distinction. J. M. Bocheński explained, “Thinkers of these schools did indeed establish chains of inference, but logic consists in studying inference, not in inferring.”²

Figure 1.1

Aristotle (384 BCE–322 BCE).

Image created by jlorenz1. This file is licensed under the Creative Commons Attribution 2.5 Generic license.

Thus, Aristotle was the first to develop formal logic, intended as a set of rules by which one could reason correctly. He produced six works on logic. In collected form, they became known as the Organon.³ Questions have been raised concerning how much of these works should really be attributed to Aristotle, as opposed to his followers.⁴

In any case, Aristotle’s examples of logical reasoning are known as syllogisms. An example that is frequently given is

All men are mortal.

Socrates is a man.

Therefore, Socrates is mortal.

However, this is not the way Aristotle actually did things. His work is represented more accurately as

If all men are mortal

1 Taken here from http://www-history.mcs.st-and.ac.uk/Quotations/Russell.html, which has many other great quotes from Russell.

2 Bocheński, J. M., A History of Formal Logic (translated and edited by Ivo Thomas), University of Notre Dame Press, Notre Dame, IN, 1961, p. 11.

3 This collection can be downloaded from https://ia601407.us.archive.org/34/items/AristotleOrganon/AristotleOrganoncollectedWorks.pdf where it appears in a form translated under the editorship of W. D. Ross. Other works of Aristotle are included in this pdf as well.

4 Bocheński, J. M., A History of Formal Logic (translated and edited by Ivo Thomas), University of Notre Dame Press, Notre Dame, IN, 1961, pp. 40–41.

and all Greeks are men,

then all Greeks are mortal.⁵

But he did much more than demonstrate such commonsensical (to us anyway) arguments. He looked at things abstractly. This process seems to have begun with abbreviations. Using single letters to stand for men, mortal, and Greek, we can make the leap to treating these letters as variables.⁶ A pair of examples follow:⁷

1. Since if A (is predicated) of all B, and B of all C, A must be predicated of all C.
2. Similarly too if A (is predicated) of no B, and B of all C, it is necessary that A will belong to no C.

These are less clear (as often happens when things become more abstract), but much more general. Replacing the variables in 1) with specific choices gives⁸

If animal belongs to all man

and man belongs to all Greek,

then animal belongs to all Greek.

5 Taken here from http://planetmath.org/aristotelianlogic.

6 Bocheński, J. M., A History of Formal Logic (translated and edited by Ivo Thomas), University of Notre Dame Press, Notre Dame, IN, 1961, p. 42.

7 Taken from Bocheński, J. M., A History of Formal Logic (translated and edited by Ivo Thomas), University of Notre Dame Press, Notre Dame, IN, 1961, p. 64.

8 Taken from Bocheński, J. M., A History of Formal Logic (translated and edited by Ivo Thomas), University of Notre Dame Press, Notre Dame, IN, 1961, p. 66.

For 2), we have an example⁹ given by

If stone belongs to no man

and man belongs to all Greek,

then stone belongs to no Greek.

The abstraction from specific examples to the general syllogisms stated by Aristotle was a very important step. He also tried to build his logic on an axiomatic foundation. That is, he found that some syllogisms followed logically from others.

This also happens in every area of mathematics. That is, there are some statements that are deductive consequences, by specified rules of inference, of others. Wouldn’t it be nice if we could f...