Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 1/refcontrol

Mathematics for applied sciences

Mathematics has a multitude of applications. E.g., it helps to find a suitable room for a given audience, by comparing the number of seats with the number of persons. Both sets are counted with the natural numbers and mathematics provides methods to compare the two numbers directly, so that one does not have to try several rooms with the people. Mathematics helps to balance monthly debits and credits into balance by computing from the monthly income how much one may spend per day.

Mathematics has applications in the technical area, in image processing, in medicine, in data transmission and data storage, in the design of search algorithms, in solving logistic problems, in data science, machine learning, artificial intelligence. Here, we are mainly interested in applications of mathematics in other sciences. Mathematics is used in many different sciences, in the natural sciences like physics, chemistry, biology, in computer sciences, it is used in the form of statistics in psychology, sociology and history, as geometric device in cartography, in economics. Mathematics provides a language to describe empirical incidents in a quantitative way, it helps modelling theories about observable phenomena. Using techniques like interpolation and extrapolation, it allows giving predictions like for the weather or the development of the population on earth.

In these different areas, a huge variety of mathematical methods is applied, which differs strongly in its complexity, in terms of the mathematical structure and in terms of the application. E.g., statistics is a complex mathematical discipline, but in many sciences it is used (for good reason) directly as a black box without deeper reflection, for example to evaluate the reliability of a psychological test. On the opposite side, in (theoretical) physics there is such a tight connection with mathematics so that the theories in physics can not even exist without the mathematics and that a large portion of mathematics originates from physics. Regarding differential equations, one might be only interested in their numerics (concrete approximating techniques of computations), or whether they are adequate for a certain problem, or for the qualitative behavior of a solution. So mathematics is used in a broad range regarding depth in the sciences.

However, for most mathematical applications, it seems fair to say that an appropriate and reasonable use of mathematics requires a profound mathematical education, which can not be acquired in the context of concrete problems alone. It is the target of this course to build a mathematical base in order to persist in several scientific contexts.

This goal explains to a large extent the content of this course. We want to give a solid foundation of mathematics, how it works, its principles, its schemes of argumentation, its conceptual framework. Contrary to this, concrete applications, as they occur in the sciences in the end, will play a minor role. This rests on the fact that realistic applications need a closer knowledge of the sciences themselves. They decide what they think is a realistic scenario and a suitable modelling. Mathematics provides suitable devices which may be used by the sciences.

It might be appropriate to explain how and why mathematics taught at school differs from mathematics of a university. The short answer is that on the university level, mathematics is pursued as a science. From that perspective, it is reasonable to consider more general the question of what scientific activity means. An important observation is that scientific statements need a scientific justification (reasoning, argumentation, proof). So let us consider what argumentation means in the scientific and in the mathematical context.

Mathematical reasoning

In a mathematical argumentation, one tries to establish the truth of a claim by only using accepted principles. Typically, there is an audience which one would like to convince about the claim. Argumentation exists in many different contexts, in sciences, in politics, in personal relationships. There exist principles and patterns of argumentation which are typical for each context. In the political context, one invokes in general widely accepted principles like human rights, the constitution, the people and what they want, in order to enforce political decisions. The experience shows that the arguments used there are not good enough to convince everybody, and that also the interests of specific groups are represented.

Also in a mathematical argumentation one tries to establish the truth of a claim (that a computation procedure is correct, that a model is appropriate). The tools used, the strictness of the argumentation, do also depend here on the audience, the previous knowledge and motivation, the relation (bond, trust) between the persons involved (like teacher and student).

A mathematical argumentation on the scientific level shows certain standards of argumentation. A scientific argumentation exhibits the following items.

• The strong presence of technical terms, which have to be defined and have to be used according to their definitions.
• The existence of quite few basic principles.
• The use of logic to conclude new knowledge.
• The free usage of knowledge, which has been already established in the sciences.
• The free accessibility and verifiability of the results.
• The claim that the validity of the knowledge is independent of subjective wishes and values,
that it is timeless and independent of culture.

In a mathematical argumentation, these items are in particular patent. A strong hint for this is that there exists even a specific term for a mathematical argumentation: proof. A proven mathematical claim is called a theorem (or lemma or corollary).

• All mathematical concepts are defined precisely using only concepts which have already been defined. The definitions are built in such a way that every reasonable mathematical object either satisfies this concept or not, independent of whether we are always able to decide this.
• Mathematics is constructed
(since about 130 years) on sets. It is organized in an axiomatic-logical way, but motivated by phenomena in the real world.
• Logic is the main tool of mathematics. There exists
(in principle) a complete list of allowed conclusions in propositional logic and in predicate logic.
• Proven mathematical statements
(theorems) are used again and again. This means that for a systematic account (like a lecture or a book) of an area of mathematics, the foundational material comes first and on this foundation more complex statements are built. If a theorem already proven is used somewhere, then we do not reflect about this theorem again, we only check whether in the current situation all conditions are fulfilled which the theorem requires.
• Mathematics is published in journals and books, it is taught in lectures, it is available on the World Wide Web and in libraries.
• Mathematics is developed in a worldwide community.

The students of a university are supposed to learn a specific science, its basic concepts, its structure, its working principles, its most important results and historic landmarks. They are supposed to gain a broad scientific competence in the chosen field. Here, we encounter three natural questions: is science difficult? Are academic studies difficult? Is this course difficult? The rough answer is yes, quite. For this course one can say the following: If the goal is to understand the content of the course profoundly in its entirety, to be able to solve all exercises and to always have an overview, then it is quite difficult and requires a huge mathematical talent. However, if the primary goal is to pass the course, then the difficulties lie rather on the organizational side: to work continuously on the subject, recognize important themes and typical questions, to find motivation, to find like-minded persons, recognize deficits and react in a suitable way (in particular realize that strategies which were sufficient at school might not be sufficient anymore).

Propositions

We have mentioned already the importance of propositions and the logic relating them in sciences in general and in mathematics. We are going to make this more precisely.

A proposition is a structure of language (a sentence) that can be either true or false. It might be the case that it is difficult to decide whether the proposition is true or false. The important thing is that the predicates true and false are reasonable predicates for the structure due to its syntactic and semantic form.

The condition that the meaning of a sentence is completely clear is usually not fulfilled by statements of the natural language. Let's have a look at the sentence

This horse is fast.

On one hand, we do not have any information about what horse we are talking, and the validity of the statement depends probably on the horse. On the other hand, the meaning of "fast“ is not so clear that even if we knew about which horse we are talking, it is difficult to agree whether we want to consider it as rather fast or not. Further examples are

Martians are green.

I eat my hat.

Heinz Ngolo and Mustafa Müller are friends.

In a natural language, we have the possibility to produce a situation (by adding information, considering the context, making agreements) where one can clarify the meaning of the statement. In logic and in mathematics however this kind of solution is not allowed. Instead, the meaning of the proposition has to be clear from the terms used in the proposition alone, and these terms have to be defined in advance. Some mathematical statements are (maybe true, maybe false)

${}5>3$.

${}5<3$.

5 is a natural number.

We have ${}7+5=13$.

Prime numbers are odd.

If we understand these statements and in particular the concepts and symbols involved, then we recognize that we are dealing with a proposition that may be true or false, and this holds independently whether we know its validity. Whether a structure of language is a proposition does not depend from our knowledge whether it is true or false, nor on the effort necessary to decide whether it is true or false. In the following we present some mathematical objects which are not propositions:

5

5+11

The set of prime numbers

${}A\cap B$

A sum of five squares

${}\int _{a}^{b}f(t)dt$.

Instead of dealing with concrete propositions, in the following we take up the position that a proposition is just a propositional variable ${}p$ which may have the logical value (truth value) true or false. We are then interested on how the assignment of logical values behaves when we construct new propositions from old propositions.

Logical connectives

Starting from several propositions, one can build new propositions. From the proposition

I eat my hat

one can build the

negative proposition

I do not eat my hat,

from the two propositions

Martians are green

and

I eat my hat

one can produce the following new propositions.

Martians are green and I eat my hat

Martians are green or I eat my hat

If Martians are green, then I eat my hat

If it is not true that Martians are green, then I eat my hat

If Martians are green, then I do not eat my hat

If it is not true that Martians are green, then I do not eat my hat

Martians are green if and only if I eat my hat

Here, the two propositions involved are not changed (maybe some slight changes due to grammar), they are just brought together into a new logical relationship. Such a logical combination (operation) is characterized by the fact that its logical value is determined by the logical values of the propositions involved and the meaning of the logical combinations (in propositional logic, these are called logical connectives). The proposition

Martians are green and I do not eat my hat

is true if and only if both propositions are true. This is the logical meaning of the and-connective. A further connection between the two statements is not necessary.

To contrast, let us consider a proposition like

The green Martians eat hats

Here a completely new proposition arises, where just some words or predicates of the two propositions occur, but its logical value can not be deduced from the given propositions.

We have a logical connective of propositions if the logical value of the whole proposition can be deduced from the logical values of the propositions involved. The logical connectives determine how the logical value of the compound proposition has to be computed from the logical values of the propositions involved.

Variables and logical connectives

In order to understand the dependence of a compound proposition on the truth values of the propositions involved and the logical connectives, it is useful to work with propositional variables and to represent the connectives by symbols. For a proposition, we just write

$p,q,\ldots ,$

and we are not interested in the content of ${}p$, only in the possible truth values (or truth assignments) of ${}p$, which we denote by ${}t$ (true) or ${}f$ (false) (sometimes we use the truth values ${}1$ and ${}0$). In the negation, the truth values are interchanged, which can be expressed by the simple truth table:

Negation
${}p$ ${}\neg p$ t f f t

In a given concrete proposition, there are several possibilities to express the negation. To negate the statement "I eat my hat“, we can say:

I do not eat my hat.

It is not the case that I eat my hat.

It is not true that I eat my hat.

The negation applies to a single proposition, it is a monadic operator. Let's have a look at operators which apply to more than one proposition, typically a binary operator. For a binary operation, which applies to two propositions, there are all in all four possible combinations of the truth values, so that every logical connective is determined on how it acts on these combinations. Therefore, there are ${}16$ logical connectives, the most important are the following.

The logical conjunction is the and-connective. It yields true if and only if both propositions involved are true; it is false as soon as one of the propositions involved is false. The truth table of the conjunction is as follows.

Conjunction
${}p$ ${}q$ ${}p\wedge q$ t t t t f f f t f f f f

The logical disjunction is the (inclusive) or-connective. It is true if at least one of the propositions involved is true, it is in particular also true if both statements are true. It is only false in the case when both propositions are false. It is clear that the conjunction and the disjunction are symmetric with respect to the propositions involved.

Disjunction
${}p$ ${}q$ ${}p\vee q$ t t t t f t f t t f f f

The implication is the most important operation in mathematics. Mathematical theorems do usually have the form of a (nested) implication. Examples are (see Corollary 6.6 and Lemma 9.10 )

If a polynomial has degree ${}d$, then it has at most ${}d$ zeroes.

If a sequence converges, then it is bounded.

The logical content of an implication is that the validity of a condition ensures the validity of a conclusion. An implication is expressed by saying: "if ${}p$ is true, then also ${}q$ is true“ (or short: if ${}p$, then ${}q$). Its truth condition is that if ${}p$ has truth value true, then ${}q$ has also truth value true. This is satisfied whenever ${}p$ is false or if ${}q$ is true. Its truth table is therefore

Implication
${}p$ ${}q$ ${}p\rightarrow q$ t t t t f f f t t f f t

For an implication, the two propositions involved do not play the same role, the implications ${}p\rightarrow q$ and ${}q\rightarrow p$ are different. An implication has a certain direction. In the general usage and also in mathematics, implications are used when the premise is the "reason“ for the conclusion, when the implication expresses a causal connection. But this interpretation does not play any role in the context of propositional logic.

If both implications ${}p\rightarrow q$ and ${}q\rightarrow p$ hold, then we express this by saying "${}p$ is true if and only if ${}q$ is true“. This is called an equivalence of the two propositions, the truth table is

Equivalence
${}p$ ${}q$ ${}p\leftrightarrow q$ t t t t f f f t f f f t

Examples for a mathematical equivalence are:

A natural number ${}n$ is even if and only if in its decimal expansion, the last digit is ${}0,2,4,6$ or ${}8$.

A triangle is rectangular if and only if there exists a side whose square equals the sum of the squares of the other sides.

The direction from left to right in the second statement is the Pythagorean theorem, but the reverse direction is also true. Caution: There are certain contexts where equivalences are formulated as implications. This usually holds for rewards and penalties but also within mathematical definitions. If one says: "if you behave well today, then we go tomorrow to the zoo“, then one usually means that we only go to the zoo in case you behave well. Mathematical definitions like "a natural number is called even if it is a multiple of ${}2$“ are to be understood as if and only if.

Using negation, it is possible to express every logical connective by the given connectives, and not even all of them are necessary. E.g., one can express the conjunction (and also the implication and the equivalence) with the help of the disjunction, the truth table

Conjunction as disjunction
${}p$ ${}q$ ${}\neg (\neg p\vee \neg q)$ t t t t f f f t f f f f

shows that the truth function of ${}\neg (\neg p\vee \neg q)$ and the truth function of ${}p\wedge q$ coincide. Therefore, these expressions are logically equivalent. In such a simple expression it is easy to compute the logical values and hence to show the equivalence. For more complicated, deeply nested expressions it is useful to compute depending on the truth values of the propositions involved the truth values of all intermediate expressions. In the given example, this would yield the table

Conjunction as disjunction
${}p$ ${}q$ ${}\neg p$ ${}\neg q$ ${}\neg p\vee \neg q$ ${}\neg (\neg p\vee \neg q)$ t t f f f t t f f t t f f t t f t f f f t t t f

It is also possible to use more propositional variables instead of two and to get by nested connectives many new propositions. The truth assignments for the compound propositions may also be expressed in larger truth tables.

Tautologies

For every single proposition and compound propositions every truth value is allowed, and the truth values of the compound propositions are determined by the truth assignments of the single propositions and the truth rules of the connectives. So, depending on the truth assignments, all statements might be true or false. However, those statements are particularly interesting which produce for every single assignment always the value true. Such propositions are called tautologies. They are important in mathematics as they represent allowed conclusions as they appear in proofs quite often. E.g., if one has already proven the statements ${}p$ and ${}p\rightarrow q$, where ${}p$ and ${}q$ represent concrete mathematical statements, then one can deduce the validity of ${}q$. In propositional logic, this deduction is represented by the tautology

$(p\wedge (p\rightarrow q))\rightarrow q.$

As mentioned, a tautology is characterized by the constant logical value true. The proof that a certain proposition is a tautology is easiest done with the help of a truth table.

Modus ponens
${}p$ ${}q$ ${}p\rightarrow q$ ${}p\wedge (p\rightarrow q)$ ${}(p\wedge (p\rightarrow q))\rightarrow q$ t t t t t t f f f t f t t f t f f t f t

Double negation
${}p$ ${}\neg p$ ${}\neg (\neg p)$ ${}p\leftrightarrow \neg (\neg p)$ t f t t f t f t

Tertium non datur
${}p$ ${}\neg p$ ${}p\vee \neg p$ t f t f t t

The rule tertium non datur goes back to Aristotle. It says that a proposition is either true or false and that there is no third possibility. The rule above says formally only that at least one truth value must hold, but the rule before says that ${}p$ and ${}\neg p$ can not be both true, what is also called the law of noncontradiction (together this is called the principle of bivalence). The validity of these rules is dubious in colloquial statements, but in the framework of propositional logic and of mathematics they hold without any restriction. This is related to the fact that in these fields only such statements are allowed which have a unique truth value. As a principle of proof, the logical principle of bivalence occurs as proof by cases, the following tautology expresses this.

Proof by cases
${}p$ ${}q$ ${}p\rightarrow q$ ${}\neg p$ ${}\neg p\rightarrow q$ ${}((p\rightarrow q)\wedge (\neg p\rightarrow q))$ ${}((p\rightarrow q)\wedge (\neg p\rightarrow q))\rightarrow q$ t t t f t t t t f f f t f t f t t t t t t f f t t f f t

In a proof by cases, one wants to proof ${}q$, and one proves it first under the additional assumption that ${}p$ holds (case 1) and secondly under the additional assumption that ${}\neg p$ holds (case 2). One has to do two steps, but in each step one can use additional information.

Contraposition is often used in proofs, and quite often this is not made explicit. In a proof we use a pragmatic viewpoint, and sometimes it is easier to pass from ${}\neg q$ to ${}\neg p$ instead of passing from ${}p$ to ${}q$.

Contraposition
${}p$ ${}q$ ${}p\rightarrow q$ ${}\neg p$ ${}\neg q$ ${}\neg q\rightarrow \neg p$ ${}(p\rightarrow q)\leftrightarrow (\neg q\rightarrow \neg p)$ t t t f f t t t f f f t f t f t t t f t t f f t t t t t

The proof by contradiction is also a useful pattern of argumentation. One shows that a certain statement ${}p$ implies a contradiction, often of the form ${}q\wedge \neg q$, and then one infers that ${}p$ can not be true, so that ${}\neg p$ must hold.

Proof by contradiction
${}p$ ${}q$ ${}p\rightarrow q$ ${}p\rightarrow \neg q$ ${}(p\rightarrow q)\wedge (p\rightarrow \neg q)$ ${}\neg p$ ${}(p\rightarrow q)\wedge (p\rightarrow \neg q)\rightarrow \neg p$ t t t f f f t t f f t f f t f t t t t t t f f t t t t t

Footnotes

Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I | >> PDF-version of this lecture Exercise sheet for this lecture (PDF)