Here are two definitions, both taken from the internet. Definition 1: A **dog** is a domesticated carnivorous mammal that typically has a long snout, an acute sense of smell, non-retractile claws, and a barking, howling, or whining voice. Definition 2: An **encumbered asset** is one that is currently being used as security or collateral for a loan.

These two definitions illustrate an important distinction which ought to be more widely appreciated: that some definitions bring into being a *new concept*, while others merely package conveniently and concisely *what we already know*.

Each of us from an early age understands what a dog is, by having many of them pointed out to us. We learn from experience that there are many different types of dog, but they mostly all have some common characteristics that generally separate them say from other animals, typically cats. The definition of a dog given above is only a summary, short and sweet, of familiar properties of the animal.

Most of us know what an asset it. But the adjective “encumbered”, when applied to assets, is not one that is familiar to us. At some point in the history of finance someone thought up this particular concept and needed a word for it. How about *encumbered*? This might have been one of several terms proposed— borrowing a word from English with a related but different meaning, and giving it here a precise new meaning.

Let’s give a name to this distinction that I am trying to draw here. Let’s say that a definition that summarizes more concisely, or accurately, something that we already know is a **rhetorical definition**. Let’s also say that a definition that creates a new kind of object or concept by bringing together previously unconnected properties is a **conceptual definition**.

If I ask you what **love** is, you will draw upon your experience with life and the human condition, and give me a list of enough characteristics that characterize love in your view. Almost everyone would have an opinion on the worth of your definition, because we all have prior ideas about what love is, and will judge whether your definition properly captures what we already know from our a priori experience. This kind of definition is largely rhetorical.

If I ask you what a **perfect number** is, and you are a good mathematics student, you will tell me that it is a natural number which is equal to the sum of those of its divisors which are less than itself. So 6 is a perfect number since 6=1+2+3, and 28 is a perfect number since 28=1+2+4+7+14. This is not the usual colloquial meaning of *perfect*: we are just hijacking this word to bring into focus a formerly unconsidered notion (this was done by the ancient Greeks in this case). This is a conceptual definition.

In mathematics, we *prefer conceptual definitions to rhetorical ones*. When we define a concept, we want our statement of that concept to be so clear and precise that it invokes the same notion to all who hear it, even those who are unfamiliar with the idea in question. Prior experience is not required to understand conceptual definitions, except to the extent of having mastered the various technical terms involved as constituents of the definition.

We do not want that in order to properly understand a term someone needs some, perhaps implicit, prior understanding of the term. If I tell you that a **number** officially is something used for counting or measurement, you are probably not happy. While this kind of loose description is fine for everyday usage, it is not adequate in mathematics. Such a rhetorical definition is ambiguous: because it draws upon your prior loose experience with counting and measuring, and we can all see that people could view the boundaries of this definition differently from others. In mathematics we want to create fences around our concepts; our definitions ought to be precise, visible and unchanging.

If I tell you that a function is **continuous** if it varies without abrupt gaps or fractures, then you recognize that I am not stepping up to the plate mathematically speaking. This is a rhetorical definition: it relies on some prior understanding of notions that are loosely intertwined with the very concept we are attempting to attempting to frame.

And now we come to the painful reality: **modern mathematics is full of rhetorical definitions.** Of concepts such as: *number, function, variable, set, sequence, real number, formula, statement, topological space, continuity, variety, manifold, group, field, ring, *and* category*. These notions in modern mathematics rest on definitions that are mostly unintelligible to the uninitiated. These definitions implicitly assume familiarity with the topic in question.

The standard treatment in undergrad courses shows you first lots of examples. Then after enough of these have been digested, you get a “definition” that captures enough aspects of the concept that we feel it characterises the examples we have come to learn. The cumulative effect is that you have been led to believe you “know” what the concept is, but the reality is something else. This becomes clear quickly when you are presented with non standard examples that fall outside the comfortable bounds of the text books.

This is a big barrier to the dissemination of mathematical knowledge. While modern books and articles give the appearance of precision and completeness, this is often a ruse: implicitly the reader is assumed to have gained some experience with the the topic from another source. There is a big difference between a layout of a topic and a summary of that topic. An excellent example is the treatment of real numbers in introductory Calculus or Analysis texts. Have a look at how cavalierly these books just quickly gloss over the “definition”, essentially assuming that you already know what real numbers supposedly are. Didn’t you learn that way back in high school?

Understanding the rhetorical aspects of fundamental concepts in pure mathematics goes a long way to explaining why the subject is beset with logical problems. Sigh. I guess I have some work to do explaining this. But you can do some of it yourself by opening a textbook and looking up one of these terms. Ask yourself: without any examples, pictures or further explanations, does this definition stand up on its own two legs? If so, then it can claim to be a logical conceptual definition. Otherwise it is more likely a dubious rhetorical definition.

AndyThanks @njwildberger excellent post!

This topic is very relevant to all subjects in general. Personally, the modern curricula I’ve encountered seemed to favor the student who is more willing to accept rhetorical leaps. The intuitively logical student usually lacks the ability to describe the hesitance they feel to accept certain notions, and questions to explore deeper often are taken lightly. This certainly was my experience.

Thanks again for describing the issue so well.

MarkUnfortunately your argument will not convince those who are content with the rhetorical definitions. They will insist the definitions are clear, precise, and rigorous. In other articles and videos you have explained many of the the problems in detail, but to no avail where these people are concerned.

If we believe we understand a given concept, we humans generally don’t like to change our minds. We need a heck of a lot of convincing, especially if we are to reject a mainstream point of view.

I think one possible way to tackle this is to start all over again with mathematics fundamentals. The subject of mathematics itself needs a clear definition. Conceptual definitions should then be provided for the fundamental building blocks, like ‘numbers’.

This approach should bring so much clarity into the field that it should shed light on today’s fundamental anomalies, such as should zero be different to other numbers. For example, the operations of multiplying by zero and dividing by zero can each cause problems. Does zero really have no multiplicative inverse and should zero to the power zero have different values in different scenarios?

Maybe a new approach would result in zero being treated exactly the same as all other numbers. Without a clear precise and useable definition for ‘number’ we are left not knowing if or why zero is different to other numbers. Personally I do not believe it should be treated differently (for example, I have my own algorithm for division where 0 divided by 0 equals 1).

Rather than idiots like me trying to re-build mathematics, it should be down to people like your good self, Professor Wildberger, ideally as part of a huge (& well funded) project. After all, this is important stuff as it underpins a lot of physics and engineering, so its not just maths that will be affected.