Talk:Mathematics/Conceptions of mathematics

Conceptions of mathematics

• This pages collects conceptions of mathematics among various groups of people.
• Use sourced quotes.
• If there is ambiguity about placement, take a guess.
• This is not an attempt at categorization (WP already does that nicely), but merely organization.

Among Mathematicians

Gauss

What Gauss said.

Courant and Robbins

1. Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science. (Courant and Robbins, p. xv) (This is the complete first paragraph.)
2. Yet as a science in the modern sense mathematics only emerges later, on Greek soil, in the fifth and fourth centuries B.C.
   (Courant and Robbins,p. xv)
3. A serious threat to the very life of science is implied in the assertion that mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician. If this description were accurate, mathematics could not attract any intelligent person. It would be a game with definitions, rules, and syllogisms, without motive or goal. (Courant and Robbins, p. xvii)
4. Courant, Richard (1941). What is Mathematics? An Elementary Approach to Ideas and Methods. London: Oxford University Press. {{cite book}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)

Among Scientists

Einstein

1. Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature. (Obituary for Emmy Noether (1935))

Among Philosophers

Popper

What Popper said.

Lakatos

What Lakatos said.

Among Children

Patrick Bossert

What Patrick Bossert said.

In Popular Culture

"Good Will Hunting" (movie)

The movie Good Will Hunting is an example of mathematics in popular culture. The use of low-resolution images is OK, according to the copyright boxes for these images.

The movie is notable because:

• Will Hunting, the main character, is a mathematical prodigy who works as a janitor at MIT.
• Professor Lambeau is a math professor at MIT, a combinatorialist, and a Fields Medalist.
• Will solves a graduate level problem overnight.
• The theorem that Professor Lambeau finishes writing on the classroom chalkboard just after we first see him is Parseval's theorem from Fourier analysis.
• Prof. Lambeau tells his class that there is "an advanced Fourier system on the main hallway chalkboard."
• The problem that is seen on the hallway chalkboard, and the first problem that Will solves, is a second year problem in algebraic graph theory.

NB:

• The Good Will Hunting article refers to the problem as an equation: "...wondering who could have solved the equation."
• At another point the article reads: "...astonished assistant staring at the correctly solved theorem."
• The Parseval's theorem article does not mention the movie.

Quotes

• Prof. Lambeau: "I know many of you had this as undergraduates, but it won't hurt to brush up." (to his class)
• Prof. Lambeau: "I also put an advanced Fourier system on the main hallway chalkboard." (first problem Will solves)
• Prof. Lambeau: "There is a problem on the board right now that took us more than two years to prove." (second problem Will solves)

References

• Mathematics in the movie "Good Will Hunting"
• The Mathematics in the Cinema Movie "Good Will Hunting"

In Encyclopaedias

• "the science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects. It deals with logical reasoning and quantitative calculation, and its development has involved an increasing degree of idealization and abstraction of its subject matter." Britannica.com
• "Mathematics, a way of describing relationships between numbers and other measurable quantities. Mathematics can express simple equations as well as interactions among the smallest particles and the farthest objects in the known universe. Mathematics allows scientists to communicate ideas using universally accepted terminology. It is truly the language of science." MSN Encarta
• Originally, the collective name for geometry, arithmetic, and certain physical sciences (as astronomy and optics) involving geometrical reasoning. In modern use applied, (a) in a strict sense, to the abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra; and (b) in a wider sense, so as to include those branches of physical or other research which consist in the application of this abstract science to concrete data. When the word is used in its wider sense, the abstract science is distinguished as pure mathematics, and its concrete applications (e.g. in astronomy, various branches of physics, the theory of probabilities) as applied or mixed mathematics. Oxford English Dictionary, 2nd Ed.
• the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations Merriam-Webster Online Dictionary.
• the branch of science concerned with number, quantity, and space, either as abstract ideas (pure mathematics) or as applied to physics, engineering, and other subjects (applied mathematics). Compact Oxford English Dictionary.
• a science (or group of related sciences) dealing with the logic of quantity and shape and arrangement WordNet 2.0, Princeton University, 2003.
• "MATHEMATICS (Gr. paOiµarLK1), sc. TEXvn or E7rio'7-)µ17; from AecO a, "learning" or "science"), the general term for the various applications of mathematical thought, the traditional field of which is number and quantity. It has been usual to define mathematics as "the science of discrete and continuous magnitude." Even Leibnitz,' who initiated a more modern point of view, follows the tradition in thus confining the scope of mathematics properly so called, while apparently conceiving it as a department of a yet wider science of reasoning. A short ' Cf. La Logique de Leibnitz, ch. vii., by L. Couturat (Paris, 190,). consideration of some leading topics of the science will exemplify both the plausibility and inadequacy of the above definition." Encyclopaedia Britannica, 1911.
• the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Random House Unabridged Dictionary, 2006.
• The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols. American Heritage Dictionary, 4th Ed., 2000.
• "Yet as a science in the modern sense mathematics only emerges later, on Greek soil, in the fifth and fourth centuries B.C.", Richard Courant and Herbert Robbins, What is Mathematics?, 1941, p. xv.