Pure Mathematics – Where Numbers Go To Meditate

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Benjamin Boster.

This sponsored episode on Pure Mathematics is a birthday wish from Lily to Alex.

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Alex!

Pure Mathematics is the study of mathematical concepts independently of any application outside mathematics.

These concepts may originate in real-world concerns,

And the results obtained may later turn out to be useful for practical applications,

But pure mathematicians are not primarily motivated by such applications.

Instead,

The appeal is attributed to the intellectual challenge and aesthetic beauty of working out the logical consequences of basic principles.

While Pure Mathematics has existed as an activity since at least ancient Greece,

The concept was elaborated upon around the year 1900,

After the introduction of theories with counterintuitive properties,

Such as non-Euclidean geometries and Cantor's theory of infinite sets,

And the discovery of apparent paradoxes,

Such as continuous functions that are nowhere differentiable and Russell's paradox.

This introduced the need to renew the concept of mathematical rigor and rewrite all mathematics accordingly with a systematic use of axiomatic methods.

This led many mathematicians to focus on mathematics for its own sake,

That is,

Pure Mathematics.

Nevertheless,

Almost all mathematical theories remained motivated by problems coming from the real world or from less abstract mathematical theories.

Also,

Many mathematical theories,

Which had seemed to be totally Pure Mathematics,

Were eventually used in applied areas,

Mainly physics and computer science.

A famous early example is Isaac Newton's demonstration that his law of universal gravitation implied that planets move in orbits that are conic sections,

Geometrical curves that had been studied in antiquity by Apollonius.

Another example is the problem of factoring large integers,

Which is the basis of the RSA cryptosystem widely used to secure internet communications.

It follows that currently the distinction between Pure and Applied Mathematics is more a philosophical point of view,

Or a mathematician's preference,

Rather than a rigid subdivision of mathematics.

Let's talk about the history a little bit,

Starting with Ancient Greece.

Ancient Greece mathematicians were among the earliest to make a distinction between Pure and Applied Mathematics.

Plato helped to create the gap between arithmetic,

Now called number theory,

And logistic,

Now called arithmetic.

Plato regarded logistic,

Arithmetic,

As appropriate for businessmen and men of war,

Who must learn the art of numbers or they will not know how to array their troops,

And arithmetic,

Number theory,

As appropriate for philosophers,

Because they have to arise out of the sea of change and lay hold of true being.

The Greek mathematician Apollonius of Perga was asked about the usefulness of some of his theorems in Book 4 of Conics,

To which he proudly asserted,

They are worthy of acceptance for the sake of the demonstrations themselves,

In the same way as we accept many other things in mathematics,

For this and for no other reason.

And since many of his results were not applicable to the science or engineering of his day,

Apollonius further argued in the preface of the 5th Book of Conics,

That the subject is one of those that seem worthy of study for their own sake.

Let's go to the 19th century.

The term itself is enshrined in the full title of the Sidlerian Chair,

Sidlerian Professor of Pure Mathematics,

Founded as a professorship in the mid-19th century.

The idea of a separate discipline of pure mathematics may have emerged at that time.

The generation of Gauss made no sweeping distinction of the kind between pure and applied.

In the following years,

Specialization and professionalization,

Particularly in the Weierstrass approach to mathematical analysis,

Started to make a rift more apparent.

At the start of the 20th century,

Mathematicians took up the axiomatic method,

Strongly influenced by David Hilbert's example.

The logical formulation of pure mathematics suggested by Bertrand Russell,

In terms of a quantifier structure of propositions,

Seemed more and more plausible,

As large parts of mathematics became axiomatized and thus subject to the simple criteria of rigorous proof.

Pure mathematics,

According to a view that can be ascribed to the Bourbaki group,

Is what is proved.

Pure mathematician became a recognized vocation,

Achievable through training.

The case was made that pure mathematics is useful in engineering education.

There is a training in habits of thought,

Points of view,

And intellectual comprehension of ordinary engineering problems,

Which only the study of higher mathematics can give.

One central concept in pure mathematics is the idea of generality.

Pure mathematics often exhibits a trend towards increased generality.

Uses and advantages of generality include the following.

Generalizing theorems or mathematical structures can lead to deeper understanding of the original theorems or structures.

Generality can simplify the presentation of material,

Resulting in shorter proofs or arguments that are easier to follow.

One can use generality to avoid duplication of effort,

Proving a general result instead of having to prove separate cases independently,

Or using results from other areas of mathematics.

Generality can facilitate connections between different branches of mathematics.

Category theory is one area of mathematics dedicated to exploring this commonality of structure,

As it plays out in some areas of math.

Generality's impact on intuition is both dependent on the subject and a matter of personal preference or learning style.

Often generality is seen as a hindrance to intuition,

Although it can certainly function as an aid to it,

Especially when it provides analogies to material for which one already has good intuition.

As a prime example of generality,

The Erlangen program involved an expansion of geometry to accommodate non-Euclidean geometries as well as the field of topology and other forms of geometry,

By viewing geometry as the study of a space together with a group of transformations.

The study of numbers,

Called algebra at the beginning undergraduate level,

Extends to abstract algebra at a more advanced level,

And the study of functions,

Called calculus at the college freshman level,

Becomes mathematical analysis and functional analysis at a more advanced level.

Each of these branches of more abstract mathematics have many sub-specialties,

And there are in fact many connections between pure mathematics and applied mathematics disciplines.

A steep rise in abstraction was seen mid-20th century.

In practice,

However,

These developments led to a sharp divergence from physics,

Particularly from 1950 to 1983.

Later this was criticized,

For example by Vladimir Arnold,

As too much Hilbert,

Not enough Poincaré.

The point does not yet seem to be settled,

In that string theory pulls one way,

While discrete mathematics pulls back towards proof as central.

Mathematicians have always had differing opinions regarding the distinction between pure and applied mathematics.

One of the most famous,

But perhaps misunderstood,

Modern examples of this debate can be found in G.

H.

Hardy's 1940 essay,

A Mathematician's Apology.

It is widely believed that Hardy considered applied mathematics to be ugly and dull.

Although it is true that Hardy preferred pure mathematics,

Which he often compared to painting and poetry,

Hardy saw the distinction between pure and applied mathematics to be simply that applied mathematics sought to express physical truths in a mathematical framework,

Whereas pure mathematics expressed truths that were independent of the physical world.

Hardy made a separate distinction in mathematics between what he called real mathematics,

Which has permanent aesthetic value,

And the dull and elementary parts of mathematics that have practical use.

Hardy considered some physicists such as Einstein and Dirac to be among the real mathematicians,

But at the time he was writing his Apology,

He considered general relativity and quantum mechanics to be useless,

Which allowed him to hold the opinion that only dull mathematics was useful.

Moreover,

Hardy briefly admitted that,

Just as the application of matrix theory and group theory to physics had come unexpectedly,

A time may come where some kinds of beautiful,

Real mathematics may be useful as well.

Another insightful view is offered by American mathematician Andy Magid.

I've always thought that a good model here could be drawn from ring theory.

In that subject,

One has the sub-areas of cumulative ring theory and non-cumulative ring theory.

An uninformed observer might think that these represent a dichotomy,

But in fact,

The latter subsumes the former.

A non-cumulative ring is a not necessarily commutative ring.

If we use similar conventions,

Then we could refer to applied mathematics and non-applied mathematics,

Whereby the latter we mean not necessarily applied mathematics.

Friedrich Engels argued in his 1878 book,

Antiduring,

That it is not at all true that in pure mathematics the mind deals only with its own creations and imaginations.

The concepts of number and figure have not been invented from any source other than the world of reality.

He further argued that before one came upon the idea of deducing the form of a cylinder from a rotation of a rectangle about one of its sides,

A number of real rectangles and cylinders,

However imperfect in form,

Must have been examined.

Like all other sciences,

Mathematics arose out of the needs of men,

But as in every department of thought,

At a certain stage of development,

The laws which were abstracted from the real world become divorced from the real world and are set up against it as something independent,

As laws coming from outside to which the world has to conform.

Since we talked about pure mathematics,

Let's now talk about applied mathematics.

Applied mathematics is the application of mathematical methods by different fields,

Such as physics,

Engineering,

Medicine,

Biology,

Finance,

Business,

Computer science,

And industry.

Thus,

Applied mathematics is a combination of mathematical science and specialized knowledge.

The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.

In the past,

Practical applications have motivated the development of mathematical theories,

Which then became the subject of study in pure mathematics,

Where abstract concepts are studied for their own sake.

The activity of applied mathematics is thus intimately connected with research in pure mathematics.

Historically,

Applied mathematics consisted principally of applied analysis,

Most notably differential equations,

Approximation theory broadly construed to include representations,

Asymptotic methods,

Variational methods,

And numerical analysis,

And applied probability.

These areas of mathematics related directly to the development of Newtonian physics,

And in fact,

The distinction between mathematicians and physicists was not sharply drawn before the mid-19th century.

This history left a pedagogical legacy in the United States.

Until the early 20th century,

Subjects such as classical mechanics were often taught in applied mathematics departments at American universities rather than in physics departments,

And fluid mechanics may still be taught in applied mathematics departments.

Engineering and computer science departments have traditionally made use of applied mathematics.

As time passed,

Applied mathematics grew alongside the advancement of science and technology.

With the advent of modern times,

The application of mathematics in fields such as science,

Economics,

Technology,

And more became deeper and more timely.

The development of computers and other technologies enabled a more detailed study and application of mathematical concepts in various fields.

Today,

Applied mathematics continues to be crucial for societal and technological advancement.

It guides the development of new technologies,

Economic progress,

And addresses challenges in various scientific fields and industries.

The history of applied mathematics continually demonstrates the importance of mathematics in human progress.

Today,

The term applied mathematics is used in a broader sense.

It includes the classical areas noted above,

As well as other areas that have become increasingly important in applications.

Even fields such as number theory that are a part of pure mathematics are now important in applications such as cryptography,

Though they are not generally considered to be part of the field of applied mathematics per se.

There is no consensus as to what the various branches of applied mathematics are.

Such categorizations are made difficult by the way mathematics and science change over time,

And also by the way universities organize departments,

Courses,

And degrees.

Many mathematicians distinguish between applied mathematics,

Which is concerned with mathematical methods,

And the applications of mathematics within science and engineering.

A biologist using a population model and applying known mathematics would not be doing applied mathematics,

But rather using it.

However,

Mathematical biologists have posed problems that have stimulated the growth of pure mathematics.

Mathematicians such as Poincaré and Arnold deny the existence of applied mathematics and claim that there are only applications of mathematics.

Similarly,

Non-mathematicians blend applied mathematics and applications of mathematics.

The use and development of mathematics to solve industrial problems is called industrial mathematics.

The success of modern numerical mathematical methods and software has led to the emergence of computational mathematics,

Computational science,

And computational engineering,

Which use high-performance computing for the simulation of phenomena and the solution of problems in the sciences and engineering.

These are often considered interdisciplinary.

Sometimes the term applicable mathematics is used to distinguish between the traditional applied mathematics that developed alongside physics and the many areas of mathematics that are applicable to real-world problems today,

Although there is no consensus as to a precise definition.

Mathematicians often distinguish between applied mathematics,

On the one hand,

And the application of mathematics or applicable mathematics both within and outside of science and engineering on the other.

Some mathematicians emphasize the term applicable mathematics to separate or delineate the traditional applied areas from new applications arising from fields that were previously seen as pure mathematics.

For example,

From this viewpoint,

An ecologist or geographer using population models and applying known mathematics would not be doing applied but rather applicable mathematics.

Even fields such as number theory that are part of pure mathematics are now important in applications such as cryptography,

Though they are not generally considered to be part of the field of applied mathematics per se.

Such descriptions can lead to applicable mathematics being seen as a collection of mathematical methods such as real analysis,

Linear algebra,

Mathematical modeling,

Optimization,

Combinatorics,

Probability,

And statistics,

Which are useful in areas outside traditional mathematics and not specific to mathematical physics.

Other authors prefer describing applicable mathematics as a union of new mathematical applications with the traditional fields of applied mathematics.

With this outlook,

The terms applied mathematics and applicable mathematics are thus interchangeable.

Historically,

Mathematics was most important in the natural sciences and engineering.

However,

Since World War II,

Fields outside the physical sciences have spawned the creation of new areas of mathematics,

Such as game theory and social choice theory,

Which grew out of economic considerations.

Further,

The utilization and development of mathematical methods expanded into other areas,

Leading to the creation of new fields such as mathematical finance and data science.

The advent of the computer has enabled new applications,

Studying and using the new computer technology itself,

Computer science,

To study problems arising in other areas of science,

Computational science,

As well as the mathematics of computation,

For example,

Theoretical computer science,

Computer algebra,

Numerical analysis.

Statistics is probably the most widespread mathematical science used in the social sciences.

Academic institutions are not consistent in the way they group and label courses,

Programs,

And degrees in applied mathematics.

At some schools,

There is a single mathematics department,

Whereas others have separate departments for applied mathematics and pure mathematics.

It is very common for statistics departments to be separated at schools with graduate programs,

But many undergraduate-only institutions include statistics under the mathematics department.

Many applied mathematics programs,

As opposed to departments,

Consist primarily of cross-listed courses and jointly appointed faculty and departments representing applications.

Some Ph.

D.

Programs in applied mathematics require little or no coursework outside mathematics,

While others require substantial coursework in a specific field.

In some respects,

This difference reflects the distinction between application of mathematics and applied mathematics.

Some universities in the UK host departments of applied mathematics and theoretical physics,

But it is now much less common to have separate departments of pure and applied mathematics.

A notable exception to this is the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge,

Housing the Lucasian Professor of Mathematics,

Whose past holders include Isaac Newton,

Charles Babbage,

James Lighthill,

Paul Dirac,

And Stephen Hawking.

Schools with separate applied mathematics departments range from Brown University,

Which has a large division of applied mathematics that offers degrees through the doctorate,

To Santa Clara University,

Which offers only the MS in applied mathematics.

Research universities dividing their mathematics department into pure and applied sections include MIT.

Students in this program also learn another skill,

Computer science,

Engineering,

Physics,

Pure math,

Etc.

,

To supplement their applied math skills.

Applied mathematics is associated with the following mathematical sciences.

Engineering.

Mathematics is used in all branches of engineering and is subsequently developed as distinct specialties within the engineering profession.

For example,

Continuum mechanics is foundational to civil,

Mechanical,

And aerospace engineering,

With courses in solid mechanics and fluid mechanics being important components of the engineering curriculum.

Continuum mechanics is also an important branch of mathematics in its own right.

It has served as the inspiration for a vast range of difficult research questions for mathematicians involved in the analysis of partial differential equations,

Differential geometry,

And the calculus of variations.

Perhaps the most well-known mathematical problem posed by a continuum mathematical system is the question of Navier-Stokes existence and smoothness.

Prominent career mathematicians rather than engineers who have contributed to the mathematics of continuum mechanics are Clifford Truesdell,

Walter Knoll,

Andre Komolgrof,

And George Batchelor.

An essential discipline for many fields in engineering is that of control engineering.

The associated mathematical theory of this specialism is control theory,

A branch of applied mathematics that builds off the mathematics of dynamical systems.

Control theory has played a significant enabling role in modern technology,

Serving a foundational role in electrical,

Mechanical,

And aerospace engineering.

Scientific computing includes applied mathematics,

Especially numerical analysis,

Computing science,

Especially high-performance computing,

And mathematical modeling in a scientific discipline.

Computer science relies on logic,

Algebra,

Discrete mathematics,

Such as graph theory,

And combinatorics.

Operations research and management science.

Operations research and management science are often taught in faculties of engineering,

Business,

And public policy.

Statistics.

Applied mathematics has substantial overlap with the discipline of statistics.

Statistical theorists study and improve statistical procedures with mathematics,

And statistical research often raises mathematical questions.

Statistical theory relies on probability and decision theory,

And makes extensive use of scientific computing,

Analysis,

And optimization.

For the design of experiments,

Statisticians use algebra and combinatorial design.

Applied mathematicians and statisticians often work in a department of mathematical sciences,

Particularly at colleges and small universities.

Actuarial science.

This science applies probability,

Statistics,

And economic theory to assess risk in insurance,

Finance,

And other industries and professions.

Mathematical economics.

This is the application of mathematical methods to represent theories and analyze problems in economics.

The applied methods use a variety of mathematical methods usually referred to non-trivial mathematical techniques or approaches.

Mathematical economics is based on statistics,

Probability,

Mathematical programming,

As well as other computational methods,

Operations research,

Game theory,

And some methods from mathematical analysis.

In this regard,

It resembles but is distinct from financial mathematics,

Another part of applied mathematics.

According to the Mathematics Subject Classification,

MSc,

Mathematical economics falls into the Applied Mathematics,

Other Classification of Category 91.

Game theory,

Economics,

Social and behavioral sciences,

With MSc 2010 classifications for game theory at Codes 91AXX Archived 2015-04-02 at the Wayback Machine and for mathematical economics at Codes 91BXX Archived 2015-04-02 at the Wayback Machine.

Other Disciplines.

The line between applied mathematics and specific areas of application is often blurred.

Many universities teach mathematical and statistical courses outside the respective departments,

In departments and areas including business,

Engineering,

Physics,

Chemistry,

Psychology,

Biology,

Computer science,

Scientific computation,

Information theory,

And mathematical physics.

Applied Mathematics Societies.

The Society for Industrial and Applied Mathematics is an international applied mathematics organization.

As of 2024,

The society has 14,

000 individual members.

The American Mathematics Society has its Applied Mathematics Group.