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Similarly to a living thing, mathematics is a unity within an environment, yet apart from it — a compartment of a larger whole, structurally distinguishable though not functionally completely isolated from or closed to its surroundings.<ref>This phrase is borrowed from [[Life#&nbsp;&nbsp;&mdash;&nbsp;&nbsp;Cells]].</ref>
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The [[Heisenberg Uncertainty Principle|Heisenberg uncertainty principle]] for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.


==Mathematics is structurally distinguishable==
An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).
 
===Axiomatic or non-axiomatic===
 
There are two possible approaches to mathematics, called by R. Feynman "the Babylonian tradition" and "the Greek tradition". They answer differently the question, whether or not some mathematical facts are more fundamental than others, more consequential facts. The same approaches apply to any theory, mathematical or not.
 
The Babylonian (non-axiomatic) tradition treats a theory as a network whose nodes are facts and connections are derivations. If some facts are forgotten they probably can be derived from the others.
 
The Greek (axiomatic) tradition treats a theory as a tower of more consequential facts called theorems, grounded on the basis of more fundamental facts called axioms. If all theorems are forgotten they surely can be derived from the axioms. The axioms are sparse and simple, not to be forgotten.
 
===Monotonic or non-monotonic===
 
The distinction between these two approaches is closely related to the distinction between monotonic and non-monotonic logic. These answer differently the question, whether or not a fact can be retracted because of new evidence. Monotonic logic answers in the negative, non-monotonic logic answers in the affirmative.
 
Non-monotonic logic is used routinely in everyday life and research. An example: "being a bird, this animal should fly", but the bird may appear to be a penguin. Another example: "the grass is wet, therefore it rained", but the cause may appear to be a sprinkler.
 
The non-axiomatic approach is flexible. When needed, some old facts can be retracted, some new facts added, and some derivations changed accordingly. Nowadays this approach is widely used outside mathematics, and only marginally within mathematics (so-called informal mathematics).
 
The axiomatic approach is inflexible. A theorem cannot be retracted without removing (or replacing) at least one axiom, which usually has dramatic consequences for many other theorems. Nowadays this approach is widely used in mathematics.
 
The non-axiomatic approach is well suited when new evidence often comes from the outside. The axiomatic approach is well suited for a theory that advances only by extracting new consequences from an immutable list of axioms. It may seem that such a development must be dull. Surprisingly, this is an illusion. Being inflexible in some sense, an axiomatic theory can be very flexible in another sense (see below).
 
===Defined or undefined===
 
Facts are formulated via notions.
 
In the non-axiomatic approach, notions are nodes of a network whose connections are definitions. If some notions are forgotten they probably can be restored from the others.
 
Searching Google for "define:line" we get "a length without breadth or thickness". Similarly we find definitions for breadth, thickness and so on, recursively. Doing so we would get a large subnetwork; here is its small fragment:
 
*Line: a length without breadth or thickness
**Length: linear extent in space
***Linear: along a line↑
***Extent: a range of locations
****Location: point or extent↑ in space
***Space: unlimited expanse in which everything is located↑
**Breadth: the extent↑ from side to side
***Side: a surface forming part of the outside of an object
****Surface: the extended↑ two-dimensional outer boundary of a three-dimensional object
**Thickness: the dimension through an object as opposed to its length↑ or width
***Width: the extent↑ from side↑ to side↑
 
(Up arrows mean: see above.) We observe that
* circularity appears routinely; for example: line→length→linear→line;
* the definition of a single notion involves recursively a large number of other, quite remote notions.
 
Such system of notions is unsuitable for a mathematical theory. Here, circularity is disallowed, and the set of involved notions is kept reasonably small (whenever possible).
 
In the axiomatic approach, notions are a tower of defined notions, grounded on the basis of more fundamental notions called undefined primitives. If all defined notions are forgotten they surely can be restored from the undefined primitives. The undefined primitives are sparse and simple, not to be forgotten.
 
Curiously, when a non-mathematical encyclopedia contains an article on a mathematical notion, two very different "definitions" may appear, one general (informal), the other mathematical (formal).
 
From now on, in this article, "definition" means a mathematical definition (unless explicitly stated otherwise).
 
The lack of definition of a primitive notion does not mean lack of any information about this notion. Axioms provide such information, to be used in proofs. Informal (intuitive) understanding of a primitive notion is communicated in a natural language. This information cannot be used in proofs, but is instrumental when guessing what to prove, how to prove, how to apply proved theorems and, last but not least, what to postulate by axioms.
 
==Mathematics is not isolated==
 
===Computers metaphor===
 
A conceptual metaphor helps to understand one conceptual domain in terms of another. For example, the desktop metaphor treats the monitor of a computer as if it is the user's desktop, which helps to a user not accustomed to the computer. Nowadays many are more accustomed to computers than to mathematics. Thus, analogies with computers may help to understand mathematics. Such analogies are widely used below.
 
===Flexible or inflexible===
 
In 1960s a computer was an electronic monster able to read from a punch tape simple [[Instruction set architecture|instructions]] stipulated by the hardware and execute them quickly, thus performing a lot of boring calculations. Nowadays some parents complain that personal computers are too fascinating. However, without software a personal computer is only able to read (say, from a compact disk) and execute instructions stipulated by the hardware. These instructions are now as technical as before: simple arithmetical and logical operations, loops, conditional execution etc. A computer is dull, be it a monster of 1960s or a nice looking personal computer, unless programmers develop fascinating combinations (called programs) of these technical instructions.
 
For a programmer, the instruction set of a given computer is an immutable list. The programmer cannot add new elements to this list, nor modify existing elements. In this sense the instruction set is inflexible. New programs are only new combinations of the given elements. Does it mean that program development is dull? In no way! A good instruction set is universal. It means that a competent programmer feels pretty free to implement any well-understood algorithm provided that the time permits (which is usually most problematic) and the memory and the speed of the computer are high enough (which is usually less problematic). In this sense a good instruction set is very flexible.
 
Likewise a mathematician working within a given theory faces its axioms as an immutable list. However, this list can be universal in the sense that a competent mathematician feels pretty free to express any clear mathematical idea. In this sense some axiomatic systems are very flexible.
 
===Universal or specialized===
 
For a user, the software of a computer is first of all a collection of applications (games, web browsers, media players, word processors, image editors etc.) All applications function within an operating system (Windows, MacOS, Linux etc.) The operating system is a universal infrastructure behind various specialized applications. Each application deals with relevant files. The operating system maintains files in general, and catalogs (directories) containing files and possibly other catalogs.
 
Likewise, the [[set theory]] is a universal infrastructure behind various specialized mathematical theories (algebra, geometry, analysis etc.)  Each specialized mathematical theory deals with relevant objects, relations and sets. The sets theory deals with [[set]]s in general, possibly containing other sets, and reduces objects and relations to sets.
 
===Motivated or indiscriminate===
 
Monkeys could type into a computer a sequence of hardware instructions; the computer could execute them; but the result of such "programming" has almost no chance to be fascinating or useful. Fascinating computer games reflect human predilections. Useful programs reflect human needs. A computer is dull for humans unless its software reflects human life in one way or another.
 
Likewise, a theorem is of no interest for humans unless it is motivated in one way or another by human life. The motivation may be quite indirect; many theorems "only" help to prove other theorems, many are appreciated "only" for their aesthetic value, etc. But some kind of motivation is necessary. Indiscriminate stream of logical consequences of the axioms is not publishable in the mathematical literature.
 
Note that "a theorem" does not mean "a motivated theorem", "an important theorem" etc., not even "an already discovered theorem". All theorems are just an indiscriminate stream of logical consequences of the axioms, including the axioms themselves.
 
Theorems of a theory are, by definition, statements that follow from the given axioms according to the given rules (called by different authors inference rules, derivation rules, deduction rules, transformation rules).
 
===From technical to human: definitions===
 
The gap between a bare hardware and a nice application is too wide for a single jump, or even a triple jump (hardware – operating system – programming language – application). Bridging the gap is a laborious task for many programmers. They compose programs of modules, and modules of subroutines. Each subroutine reduces a bit more useful task to a bit simpler tasks. Ultimately, a useful (or even fascinating) task is reduced to the technical instructions of the bare hardware.
 
Likewise, mathematicians bridge the wide gap between useful notions (say, "ellipse" or "normal distribution") and the primitive notions by a large and complicated system of definitions. Each definition reduces a bit more useful notion to a bit simpler notions.
 
Mathematical definitions are very diverse.
 
A definition may be just an abbreviation local to a single calculation, like this: "denoting for convenience <math>x^2+x+1</math> by <math>a</math> we have...".
 
A single definition may embed a whole specialized mathematical theory into the universal set-theoretic framework, like this:
 
"A Euclidean space consists, by definition, of three sets, whose elements are called points, lines and planes respectively, and six relations, one called betweenness, three called containment and two called congruence, satisfying the following conditions: ..."
 
A definition may be given mostly for speaking more concisely, like this: "in a triangle, the altitude of a vertex is, by definition, its distance to the line through the other two vertices".
 
A definition may introduce a new revolutionary concept, like this: "the derivative of a function at a point is, by definition, the limit of the ratio..."
 
"Some mathematicians will tell you that the main aim of their research is to find the right definition, after which their whole area will be illuminated. ... For other mathematicians, the main purpose of definitions is to prove theorems..." [The Companion, pages 74-75]
 
==Principles and practices==
 
===Sharp or fuzzy; real or ideal===
 
A fair coin is tossed 1000 times; can it happen at random that heads is obtained all the 1000 times? Emile Borel, a famous mathematician, answers in the negative, with an important reservation: "Such is the sort of event which, though its ''impossibility'' may not be rationally demonstrable, is, however, so unlikely that no sensible person will hesitate to declare it actually impossible." (Borel, page 9)
 
Why `may not be rationally demonstrable'? Why does this statement remain outside the list of Borel's theorems in probability theory?
 
Mathematical truth is sharp, not fuzzy. Every mathematical statement is assumed to be either true or false (even if no one is able to decide) rather than "basically true", "true for all practical purposes" etc. If ''n'' heads out of ''n'' times is an impossible event for ''n''=1000 then there must exist the least such ''n'' (be it known or not). Say, 665 heads can occur, but 666 heads cannot. Then, assuming that 665 heads have just appeared, we see that the next time tails is guaranteed, which contradicts to the assumed memoryless behavior of the coin.
 
Another example. The number 4 can be written as a sum of units: 1+1+1+1. Can the number 2<sup>1000</sup> be written as a sum of units?  Mathematics answers in the affirmative. Indeed, if you can write, say, 2<sup>665</sup> as a sum of units, then you can do it twice (connecting the two copies by the plus sign). Complaints about limited resources, appropriate in the real world, are inappropriate in the imaginary, highly idealized mathematical universe.
 
 
==Notes==
<references />

Latest revision as of 03:25, 22 November 2023


The account of this former contributor was not re-activated after the server upgrade of March 2022.


The Heisenberg uncertainty principle for a particle does not allow a state in which the particle is simultaneously at a definite location and has also a definite momentum. Instead the particle has a range of momentum and spread in location attributable to quantum fluctuations.

An uncertainty principle applies to most of quantum mechanical operators that do not commute (specifically, to every pair of operators whose commutator is a non-zero scalar operator).

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