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.
Thanks @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.
This is a ridiculous situation in mathematics with all that usual smug claims of rigor.
Linguists are aware of the dictionary problem for about 200 years already.
Some set of words in a dictionary have to be linked outside of the dictionary to real world objects, otherwise you will
have undefined words or circular definitions.
Computer scientists know that any statement in programming language has to be translatable into some state of a machine.
And only in mathematics you are allowed to invoke some utter nonsense, e.g. “This axiomatization of arithmetics has an infinite model”, like a magic spell to guard yourself from requests about explicit real world examples.
Unfortunately 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.
If 0/0=1, what would 0/0+0/0 be, it could be 1+1=2, or (0+0)/0=0/0=1. So 1=2? It’s actually useful to have an operation that isn’t allowed, it tells us that we have put something that makes no sense into a formula, take distance/time=speed, in go zero seconds in zero meters and you have no way of knowing what speed is. Go 10 meters in zero seconds then you speed is? It’s impossible to have infinite speed in the same way that you can’t divide by zero.
A good hint that you’re about to be served a rhetorical definition is if the definitions is not given upfront, or merely glossed upfront with the assurance that things will be clarified later. There is never a reason to truck onward working with terms before their meaning is clear.
Your post sent me back to school, to recall the definition of “continuous function”.
After consulting Wikipedia, I think I can say that, when presented with the concept in high school, I was shown a number of examples, counter examples, and diagrams, then the Weierstrass epsilon–delta definition, followed by a more formal definition in terms of neighborhoods (briefly, for any neighborhood in the range of the function, there exists a neighborhood in the domain of the function “which maps to that neighborhood in the range of the function”). Then, in my college course, I was introduced to the topological space definition of continuity, “pre-image of an open set is an open set”.
The examples gave me an intuitive idea of continuity along with an appreciation of cases where it is absent (e.g., sin(x)/x, sin(1/x), 1/x, near x = 0). Then the formal definitions gave me a much more precise conception along with a sense of confidence that I had the formal equipment to use in any future case. The topological definition, being independent of metric, was very satisfying to me, albeit mind-bending.
Now to your concerns about rhetorical versus conceptual definitions and the state of pedagogy in undergraduate math courses.
I think that if a person is to gain mastery of a concept—particularly a subtle one, say, continuity of a function—then one must gradually learn that concept, possibly on many levels of understanding and from multiple points of view. Also, one must use the concept in practical circumstances, for instance, in a concrete case, or in the proof of a theorem.
The implication here is that one will rarely gain this mastery in a single course of instruction: the learning process is more like maturation, taking time and practice. Also, I submit that it is folly to think that a textbook, regardless of how well written, can ever meet all of the needs of the student. Most students will need some mentorship. And all must do the exercises, and reflect on their results. Learning entails a lot of work.
In pedagogy, one may perhaps begin with a “rhetorical definition” (I prefer to call these “examples”), and then graduate to a formal “conceptual” definition, as was my experience. Also, along the way, the student should be encouraged to use the formal definition in proofs, thereby acquiring hand-on experience.
I think that both approaches, “rhetorical” and “conceptual”, are meritorious, and even useful. I would consider using both.
In “The Green-Tao Theorem”, on January 3, 2016, you wrote: “…I reject modern topology’s reliance on infinite sets of infinite sets to establish continuity.”
Um, I had not been aware of your objections, as stated above. I hope my comments were not off the mark.
“concepts such as: number, function, variable, set, sequence, real number, formula, statement, topological space, continuity, variety, manifold, group, field, ring, and category”
Maybe I am going to make a big mistake somewhere, but I’m going to try giving a conceptual definition of some of these things.
You do have to start with something like the natural numbers just existing unless you want an infinite regress which I think is worse, it’s the problem of Münchhausen’s trilemma.
The best you can do for naturals is define “zero” as being a natural, and say that if you have a natural, there is something called the “successor” of that natural that is another natural. Then define addition and multiplication recursively, basically Peano arithmetic or something similar. “zero” and “successor”
A integer is a natural with a true/false value for positive/negative.
Rationals are then a pair of integers.
Then a variable just a placeholder that is allowed to be replaced with a natural/rational number, and the variable follows the same rules as the thing it is a placeholder for, that way a rational/interger number can replace the placeholder without any rules being broken.
A function is then a sentence of symbols we have defined which includes a variable, the input of the function is a number we replace the variable with, the output of the function is the number the sentence is now equal to.
A statement is a list of symbols where all the symbols have been defined and the rules of the symbols have been followed.
To define continuity you need to define limits.
Define “there exists an X when STATEMENT” as meaning it is true that you can find a number that replaces X that makes STATEMENT true.
“for any X, STATEMENT” means that it is false that there exists an X that makes STATEMENT false.
Take a function that takes a rational number. The “limit as a function F approaches X” is defined as a number Y which obeys this property, for any A (there exists a B such that(for any Z(if abs(X-Z)<B then abs(f(Z)-Y)<A)))
A function F is continuous at X if the limit as F approaches X is equal to F(X)
A function is Continuous if for any X, it is continuous at X.
Other than basic logic, zero, successor, truth and rule, I think these definitions are conceptual. Did I leave out any underlying assumptions. I left out the algorithms for operations on the naturals, integers and rationals because they are simple enough to define. Please point out the mistakes I have made.