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[…] Well-Defined […]

ReplyBy Math Vault | General Math

All right. After having done some heavy-duty math this semester, we have decided to slack off a bit for good and ponder on, hmm…*a different kind of mathematics* altogether.

“What?”, you ask? Well, if you are reading this, chances are that either you are non-mathematical, or you have been doing math for quite a while now. Either way, we want to ask you this: did it ever occur to you to ponder about what a **mathematician** actually is?

(Actually, there is mathematician right in front of you inside the head logo. By the way, his name is **Pythagoras **— or so do we believe.)

And before you make up your mind, here is one solid definition from the encyclopedia ** Uncyclopedia**:

## A

mathematician(not to be confused with Mathemagician) is a complicated mechanism for turning coffee into theorems (except for the rare oxygenarians). Because coffee is generally a much more useful commodity than theorems, most humans believe that mathematicians are useless. Nevertheless, they are produced in significant quantities, especially in Russia. Due to the sad fact they were constructed for this purpose, theoretic[al] probability predicts an infinitesimal chance of [them] getting laid.## […]

## A mathematician consists of a coffee uptake valve, a steam engine, and a caffeine converter, as well as a positronic brain used to generate theorems. The other parts of mathematicians are specifically (but not very well) designed to help them blend into the general population.

## Mathematicians may easily be mistaken for humans at first glance. However, even if the characteristic lack of a prehensile tail is hidden in the mathematician’s pants, it is relatively easy to deduce that a mathematician is not human. Because they are very specialized machines, mathematicians will usually forget anything they are told that doesn’t specifically relate to their function. They also tend not to be able to use their hands, except occasionally for typing and/or masturbating, because they can’t get laid any other way.

Hahaha… All right. Jokes aside. Let’s get to the real thing shall we? It turns out that by looking at how mathematicians use their terms, we can get a sense of how their mind work, how they conceive the world to be, and — most importantly — why they enjoy what they do. With that said, we present the following *glossary of non-mathematical terms* frequently used by mathematicians which, hopefully, might help us deconstruct (and construct!) some of the **stereotypes** about mathematicians (or even mathematics in general, for that matter).

For your own pleasure, we have included here *25 frequently-recurring terms* in the discourse of pure mathematics — terms which we believe warrant some clarifications:

(Note: The following can be construed as an in-depth analysis of mathematicians and their **buzzwords** — from a 3^{rd}-person objective point of view)

A mathematical object alleged to be a member of a certain class is **well-defined** when it indeed satisfies the definition of being in that class. For example, a certain function $f$ is well-defined when it adheres to all the clauses in the definition of a function. Similarly, a set is well-defined when it satisfies the **set theory axioms** which dictates what a set could possibly be (and cannot be).

Unlike the natural language, the strengths (and limitations) of mathematics ultimately relies on its objects being *unambiguously* well-defined. In fact, it is the rigidity of the definitions which pave the way for a solid foundation in higher mathematics.

A proof straight from the definitions (i.e., one which doesn’t rely on the more advanced theorems). In some cases, proving from first principles could end up being *harder* or *more tedious* than relying on well-established higher theorems. In fact, such is the case when it comes to proving the fact that $\displaystyle \lim_{n \to \infty} \frac{n+2}{n^2+7} = 0$.

A colloquial term for the more advanced theorems — in the relevant disciplines — which can be used to tackle a problem with more ease. In all honesty and with due respect, mathematicians very much regard their unanswered questions as *enemies*, problem solving as a form of *mental war*, the latter of which requires higher theorems as tools in their *arsenal*. The **non-violent war mentality** is indeed very much alive… 😉

An **elementary proof** of a claim is one that invokes only basic notions and methods, and while it is not necessarily a proof from first principles, it generally does not involve considerable amount of development into the subject matter either. On the contrary, a proof which requires heavy mathematical machineries from different fields is commonly referred to as a **deep result**, and a theorem is typically considered deep *until an elementary proof can be found*. For example, while the irrationality of $\sqrt{2}$ certainly has an elementary proof, the irrationality of $\pi$ is generally considered as a deep result requiring some serious chop in **real analysis**.

In mathematics, when one presents a proof without providing the necessary details in a *satisfactory* fashion, one can be accused of committing the act of **handwaving**.

In general, handwaving is considered as a sign of inability or unresourcefulness. However, there are cases of handwaving where it’s excusable or even encouraged. “What the hell?”, you ask? Well, Wikipedia explains:

## Competent well-intentioned researchers and professors rely on handwaving when, given a limited time, a large result must be shown and minor technical details cannot be given much attention—e.g., “It can be shown that

zis even.”

In pure mathematics, mathematicians spend a considerable amount of time doing **proof**, which — by most accounts — is a finite sequence of *sentences* (with the last one being the *conclusion*), each of which is either a **premise**, an **axiom**, or a result derived using the **rules of inference**. In the world of black-and-white reductionism, mathematicians tend to accept only the proofs which could potentially adhere to the above standard. And there is no middle ground for being partially valid either — in spite of the fact that the *standard of rigor* did increase historically over time.

As creatures whose survival depends on the effective wielding of the Sword of Deductive Logic, mathematicians tend to develop a painstaking *obsession with details* and a *distaste towards imprecise/unsophisticated claims*. Similar to the Macho culture among the *chauvinist*, mathematicians also tend to develop an *inherent bias* against **handwaving — **or any other act aiming to conceal one’s mathematical ignorance.

In *A Mathematician’s Apology*, G.H. Hardy left this quote about the nature of mathematical reality:

## I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems which we prove, and which we describe grandiloquently as our “creations,” are simply the notes of our observations.

When we say that someone proves or disproves a claim, we are also saying that s/he *discovers* the truth or falsehood of that claim. However, when that happens, it also means that it’s about time to move on, and focus on some other potentially more involved questions.

In general, mathematicians have established among themselves a well-known culture of *relentlessly asking questions *and* seeking for mental challenges,* and whereas people tend to shy away from tough problems, mathematicians tend to wrestle and struggle with them — even up to the point of developing an *obsession*. Much like any other exploration process, there are fleeting moments of great discoveries and lengthy periods of stagnation. It is hence in this sense that a mathematician’s *maddening* mindset exemplifies the **scientific spirit** in its *purest* form (pun intended).

As an anonymous mathematician mentioned in Quora:

## You are often confident that something is true long before you have an airtight proof for it (this happens especially often in geometry). The main reason is that you have a large catalogue of connections between concepts, and you can quickly intuit that if X were to be false, that would create tensions with other things you know to be true, so you are inclined to believe X is probably true to maintain the harmony of the conceptual space. It’s not so much that you can imagine the situation perfectly, but you can quickly imagine many other things that are logically connected to it.

In fact, mathematics constitutes one of the academic disciplines where the research is **democratized** from the get-go: you have an **intuition** about the validity of a claim, and given some relevant knowledge about the subject matter, you can set out to substantiate a claim — or you might even disprove yourself in the process. All of these can be carried out by *solely* exercising one’s mental faculty — no fancy machine, clinical trial, research assistant or controlled experiment needed. Hence comes the term **pure science**.

While not a very commonly used word, many mathematicians engage in the process of **theorisation** on a daily basis. What happens is that as one discovers more and more theorems, one becomes increasingly compelled to pick and choose the starting assumptions (a.k.a., **axioms**) from which the known theorems can be deduced. And much like a *Lego-block construction process*, mathematicians would exercise their **creativity** in the creation of a **mathematical theory**, which can be visualized as a *network of axioms and their consequences*. Insofar as mathematicians are concerned, a satisfying mathematical theory can be liken to a *standalone military complex* — or any other architectural design with *robust materials* and *minimal fillers*.

At the end of the day, the byproducts of a mathematician’s theorisation is their own creation — very much the same way a computer scientist is the mastermind behind their own web applications, or that a graphic designer stands tall as the creator of their work. Indeed, there is a God in each one of us. 😉

In certain kinds of proof, it is customary to make an additional assumption in the beginning of the proof, with the understanding that once the claim is proved *under that assumption*, the claim would be considered fully proved *even with the assumption waived*. In which case, the proof would usually start with a phrase of the form “**without loss of generality**, assume that condition X holds.”

To illustrate how **WLOG** can be used in a proof, let’s say that our task at hand is to prove the irrationality of $\sqrt{2}$ *by contradiction*. In which case, we can start by supposing that $\displaystyle \sqrt{2}=\frac{a}{b}$, and that *without loss of generality*, *assume that* $b>0$.

The reason why this is a valid assumption, is because if $b<0$, then we can always *rearrange* the fraction so as to make the denominator positive — hence no loss of generality here in making that assumption. Therefore, as long as we can prove the irrationality of $\sqrt{2}$ under the **special case** where $b>0$, we would have in effect proved the same claim in all of its generality.

A shorthand for “*if and only if*“. X iff Y (sometimes framed as “X is **necessary** and **sufficient** for Y”) exactly when X and Y are *logically equivalent claims*. That is. the conditions under which X holds are precisely the conditions under which Y holds. In which case, X and Y are **deductively interchangeable** even if they could have very different meaning.

A fancy way of saying “just by looking”. For example, **by inspection**, the function $ f(x)=x^2+2x+1$ has zero at $x=-1$.

In an educational setting, phrases such as “*by inspection*” or “*it can be easily shown that*” can be invoked even for claims which would demand *tedious computations*. In fact, such practices have been recognized as a cunning tactic to transfer the burden of *verification* from educators to students. Nevertheless, in the context of proof presentation, the clever use of “by inspection” can indeed shorten a proof significantly — by avoiding unnecessary clutters which could distract the learners from seeing the big picture.

In the pursuit of more powerful theorems, mathematicians tend to make a habit of kick-starting a discovery process by focusing on the *simpler cases/instances* first, and leverage their knowledge on these **examples** to make *bolder* claims as they gain more competence on the subject matter. **Generalizing** on the *behavioural patterns* of the starting examples usually translates into a gradual *loss of concreteness* in the reasoning process, and the increase in **abstraction** can render statements and concepts less and less *visualizable*.

Much like the same way a child will eventually learn to interpret the symbolic meaning of an object, a mathematical entity can also be represented in different ways under different viewpoints: a metric could have a **definitional formula** which explains what it does, and a **computational formula** which allows for *effective number-crunching*. As another example, the area of a **half-circle** of radius 5, which is usually represented *geometrically*, can be also conceived as the **integral** $\displaystyle \int_{-5}^{5} \sqrt{25-x^2}\, dx$. And lastly, a **combinatorial identity**, such as the fact that $\displaystyle \binom{n}{0} + \ldots + \binom {n}{n} = 2^n$, can be established either through **algebraic manipulations**, or through the use of **combinatorial arguments **(i.e., methods and techniques in counting).

While seemingly unnecessary and redundant, having **multiple representations** of the same concept — which can be liken to having more tools at our disposal — can greatly *facilitate* the process of problem solving, as it occurs frequently in mathematics that the effective tackling of a problem often requires having the *right kind of perspective,* and that some representation of a mathematical entity — while useful in one scenario — may not even be workable in another.

Reminiscent to an architect chasing for stronger materials for the next* *skyscraper, mathematicians are constantly in search of theorems of higher **mathematical strength**. Intuitively, the conception of mathematical strength is based upon the ability of a claim to prove other claims. By definition, a claim is **stronger** than the other when it proves the latter, but *not vice versa*.

While mathematicians work hard to produce stronger and stronger theorems, they are also very much interested in knowing about the **weakest condition** under which a theorem would hold. Why? Well, this is because if A is known to be weaker than B, then the fact that A implies a conclusion C would mean that *B automatically implies the same conclusion as well*. In which case, the statement “A implies C” would actually be *stronger* than the statement “B implies C”.

A word reserved for claims or cases which are *overly obvious* or *not worthy of much attention*. A **trivial result** could be one that follows straight from the definitions, and when a proof involves several cases, a **trivial case** could be one that takes one or two lines to finish.

On the other hand, if a theorem is said to be **non-trivial**, there is usually an understanding that it relies on a *cascade of higher results* which need to be proved in advance. As expected, the use of *euphemisms* is widespread in mathematics.

As a preamble, we would like to mention that the act of **personification** is very much rampant in the mathematical discourse. In particular, mathematicians tend to portray the qualities of an mathematical object as if they are *its own behaviours* — like a living creature of some sort. On the top of that, the need for **anthropomorphization** calls for mathematicians to expect the objects to *behave according to the rules and norms they have in mind*. In other words, mathematicians ascribe different values to different mathematical objects depending on *how well or regular they behave* (i.e., are perceived to behave). **Well-behaved** objects, such as the *zero function*, are sometimes also referred to as **nice** objects.

On the other side of the token, a **pathological **object, as the name implies, is one which possesses properties *contrary to most other objects in the same genre*. One such example would be the **Dirichlet’s function** (a.k.a. the *rational number indicator*), which is defined to be 0 for irrational numbers and 1 for rational numbers. The Dirichlet’s function is pathological in the sense that it actually never attains continuity at any point.

In the study of a certain family of mathematical objects, it often occurs for the objects defined by the simplest parameters to *deviate* significantly from the rest. In which case, we would recognize these objects as the *degenerate cases *— among the objects belonging to the family in question.

In geometry, for instance, a **point** can be considered as a degenerate case of a **circle** (with radius 0), and a **line** as a degenerate case of a **parabola** (whose the leading coefficient is 0). For a more elaborated example, notice that when the **hyperbola** defined by $x^2 – y^2 = 1$ is altered so that the right-hand side becomes 0, it degenerates into *a pair of diagonal lines*, as illustrated by the following graphs:

A colloquial term for **category theory**, a mathematical subject concerning the notion of **category** and the formalization of **mathematical structures** (e.g., groups*, *sets). Since different branches of mathematics could deal with different mathematical structures *which all satisfy the definition of a category*, category theory can be regarded as an *unifying mathematical theory* which allows for the establishment of some general results commonly shared by those different structures — across different mathematical disciplines.

Students in mathematics usually get their first taste of what category theory could be like through a first course in **abstract algebra**, and since category theory frequently talks about properties shared by a collection of objects (or properties of *a collection of a collection of objects*), it involves a high degree of abstraction and hence the term “**abstract nonsense**“.

Derived from the root “canon”, which means *standard convention* or *exemplar representative*. A **canonical proof** is one that has been accepted as the standard proof by convention (e.g., Euclid’s proof that there are infinitely many prime numbers). A **canonical definition** is generally one that’s considered the most natural to adopt (e.g., an even number as a number divisible by 2, and not *a number which has the same remainder as 100, when divided by 2*).

Being a horde of irresponsible language users, mathematicians like to exploit the colloquial concept of **smoothness** to refer to the **differentiability** of a function on a certain interval. While in some cases, smoothness can also refer to the **infinite-differentiability **or the **analyticity** of a function, it’s generally clear that if a function has a “bump” somewhere in its graph (as in what happens to the **absolute value function** at 0), then it’s definitely not considered as smooth there at all.

A function is said to **vanish** at a point when it assumes the value 0 (or *converges* to 0) at that point. The **absolute value function** right above you, for instance, vanishes at $x=0$. That is, the function value “*disappears*” at 0. In a similar spirit, a function (possibly a *multivariate* one) is said to **vanish at infinity** when it converges to 0 *as the length of points/vectors tends towards infinity*.

The pursuit of the **path of least resistance** constitutes the holy grail and an integral part in the conception of **mathematical beauty** and **elegance**. In general, mathematicians don’t contend with just one proof of a claim. In fact, they are constantly in hunt for the simplest proof — or one that involves the least amount of **higher theorems** and clumsy **mathematical machineries**.

Embedded in the concept of *elegance* and *beauty* is also a proof’s ability to provide meaningful* ***insight** into the subject matter. For this reason, it is in the spirit of mathematics to dislike breaking a proof into, say, a dozen of subcases — each involving some tedious and clumsy computations. Instead, more effort is invested into constructing a proof which — concisely and satisfactorily — explain *why* the claim must hold. In a sense, it is not surprising why mathematicians are appealed to this conception of mathematical beauty and elegance, as it is the **understanding** that makes knowledge more meaningful and memorable — and not the **mechanical computation** itself.

In addition to finding the simplest and most insightful proof, mathematicians are also constantly in search of **general results** which connect the pre-existing knowledge together, and that of course plays into the pursuit of elegance and beauty as well — It’s true. Mathematicians are very much into **unifying theories***,* the same way physicists are very much into, say, *the theory of electromagnetism* or *the theory of everything* (TOE).

As a profession which values *precision* and *accuracy*, some mathematicians do go further and make a distinction between elegance and beauty. For instance, Gian-Carlo Rota, a founding father of **modern combinatorics**, made it clear in his 1977 *The Phenomenology of Mathematical Beauty* that elegance is an attribute which applies mainly to the *presentation of proof and theory*, whereas beauty (i.e., enlightenment) is an attribute which applies to mathematical entities (e.g., axioms, theorems, theories) themselves, and as such are subject to *cultural and temporal influences*:

## There is a difference between mathematical beauty and mathematical elegance. Although one cannot strive for mathematical beauty, one can achieve elegance in the presentation of mathematics. In preparing to deliver a mathematics lecture, mathematicians often choose to stress elegance and succeed in recasting the material in a fashion that everyone will agree is elegant. Mathematical elegance has to do with the presentation of mathematics, and only tangentially does it relate to its content.

As people who spend a considerable amount of time *romanticizing* about **idealized objects** and concepts, mathematicians are highly susceptible to developing *non-sexual attraction* towards mathematical objects such as *functions*, *sets* or *geometrical figures*. In fact, it’s not rare to see certain mathematicians being fascinated by the **circle**, being obsessed with the **Euler’s Identity**, or enjoying the smoothness of the so-called **Hagoromo Fulltouch Chalk**.

An acronym for the Latin phrase “*quod erat demonstrandum”*, which can be translated to “which has to be demonstrated”. **Q.E.D.** is frequently found in mathematical publications at the end of a proof, although due to the advent of printing technology, Q.E.D. has been gradually replaced by the symbols □ and ■ (a.k.a **Halmos’ tombstone**). Perfect term to end the Glossary!

OK! We hope that the above tidbits provide you with some insight into the *culture* and *worldview* of pure mathematics. Remember, there is a mathematician in every one of us, and should you demand to be subject to more *mathematical brainwashing*, our Facebook can certainly help a bit on that front in terms of making your head feel a bit *challenged*. 🙂

And before we call it a day, we thought we would leave you with an *optimistic* remark by an anonymous mathematician from Quora:

## The quantitative and logical techniques you sharpen as a mathematician allow you to take many shortcuts that make learning other fields easier, as long as you are willing to be humble and modify those mathematical habits that are not useful in the new field.

Math Vault and its Redditbots enjoy advocating for mathematical experience through *digital publishing* and the *uncanny* use of technologies. Check out their **10-principle learning manifesto** so that you can be transformed into a fuller mathematical being too.

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(EN) – The Mathematician’s Glossary of Non-Mathematical Terms | mathvault.ca | Glossarissimo! June 23, 2015

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