Tag Archives: proof

exposition, language of math, math, proof, understanding math

Rigorous proofs

2 Comments

Rich and rigorous

When we try to understand a math statement, we visualize what the statement says using metaphors, images and kinetic feelings to feel how it is true, or to suggest that the statement is not true.

If we are convinced that it is true, we may then want to prove it. Doing that involves pitching out all the lovely pictures and metaphors and gestures and treating the mathematical objects involved in the proof as static and inert. “Static” means the object does not change. “Inert” means that it does not affect anything else. I am saying how we think about math objects for the purpose of rigorous proof. I am not saying anything about “what math objects are”.

In this post I give a detailed example of a proof of the rigorous sort.

Example

Informal statement

First, I’ll describe this example in typical spoken mathematical English. Suppose you suspect that the following statement is true:

Claim: Let $f(x)$ be a differentiable function with $f'(a)=0$.
Going from left to right, suppose the graph of $f(x)$ goes UP before $x$ reaches $a$ and then DOWN for $x$ to the right of $a$
Then $a$ has to be a local maximum of the function.

This claim is written in informal math English. Mathematicians talk like that a lot. In this example they will probably wave their hands around in swoops.

The language used is an attempt to get a feeling for the graph going up to $(a,f(a))$ and then falling away from it. It uses two different metaphors for $x\lt a$ and $x\gt a$. I suspect that most of us would want to clean that up a bit even in informal writing.

A more formal statement

Theorem: Let $f$ be a real valued differentiable function defined on an open interval $R$. Let $a$ be a number in $R$ for which $f'(a)=0$. Suppose that for all $x\in R$, $f$ increases for $x\lt a$ and decreases for $x\gt a$. Then $f(a)$ is a maximum of $f$ in $R$.

Proof

  1. By definition of derivative, \[\lim_{x\to a}\frac{f(x)-f(a)}{x-a}=0.\]
  2. By definition of limit, then for any positive $\epsilon$ there is a positive $\delta$ for which if $0\lt|x-a|\lt\delta$ then \[\left|\frac{f(x)-f(a)}{x-a}\right|\lt\epsilon.\]
  3. By requiring that $\delta\lt 1$, it follows from (2) that for any positive $\epsilon$, there is a positive $\delta$ for which if $0\lt|x-a|\lt\delta$, then $|f(x)-f(a)|\lt\epsilon$.
  4. “$f$ increases for $x\lt a$” means that if $x$ and $y$ are numbers in $R$ and $x\lt y\lt a$, then $f(x)\lt f(y)$.
  5. “$f$ decreases for $x\gt a$” means that if $x$ and $y$ are numbers in $R$ and $a\lt x\lt y$, then $f(x)\gt f(y)$.
  6. “$f(a)$ is a maximum of $f$ in $R$” means that for $x\in R$, if $x\neq a$, then $f(x)\lt f(a)$.
  7. Suppose that $x\in R$ and $x\lt a$. (The case that $x\gt a$ has a symmetric proof.)
  8. Given $\epsilon\gt0$ with $\delta$ as given by (3), choose $y\in R$ such that $x\lt y\lt a$ and $|f(y)-f(a)|\lt\epsilon$.
  9. By (4), $f(x)\lt f(y)$. So by (8), \[\begin{align*}
    f(x)-f(a)&=
    f(x)-f(y)+f(y)-f(a)\\ &\lt f(y)-f(a)\\ &\leq|f(y)-f(a)|\lt\epsilon\end{align*}\]
    so that $f(x)\lt f(a)+\epsilon$. By inserting “$-f(y)+f(y)$” into the second formula, I am “adding zero cleverly”, an example of pulling a rabbit out of a hat. Students hate that. But you have to live with it; as long as the statements following are correct, it makes a valid proof. Rabbit-out-of-a-hat doesn’t make a proof wrong, but it does make you wonder how the author thought of it. Live with it.
  10. Since (9) is true for all positive $\epsilon$, it follows that $f(x)\leq f(a)$.
  11. By the same argument as that leading up to (10), $f(\frac{x-a}{2})\leq f(a)$.
  12. Since $f(x)\lt f(\frac{x-a}{2})$, it follows that $f(x)\lt f(a)$ as required.

About the proof

This proof is intended to be a typical “rigorous” proof. I suspect it tends to be more rigorous than most mathematicians would find necessary,

Extensionality

The point about “rigor”, about insisting that the objects be static and inert, is that this causes symbols and expression to retain the same meaning throughout the text. This is one aspect of extensionality.

Of course, some of the symbols denote variables, or variable objects. This does not mean they are “varying”. I am taking this point of view: A variable refers to a math object but you don’t know what it is. Constraints such as $x\lt a$ rule out some possible values but don’t generally tell you exactly what $x$ is. There is more about this in Variable Objects

The idea in (6), for example, is that $y$ denotes a real number. You don’t know which number it is, but you do know some facts about it: $x\lt y\lt a$, $|f(y)\lt f(a)|\lt\epsilon$ and so on. Similarly you don’t know what function $f$ is, but you do know some facts about it: It is differentiable, for example, and $f'(a)=0$.

My statement that the variables aren’t “varying” means specifically that each unbound occurrence of the variable refers to the same value as any other occurrence, unless some intervening remark changes its meaning. For example, the references to $x$ in (7) through (10) refer to the same value it has in (6), and (10), in particular, constitutes a statement that the claim about $x$ is correct.

Checkability

The elimination of metaphors that lets the proof achieve rigor is part of a plan in the back of the mind of at least some mathematicians who write proofs. The idea is that the proof be totally checkable:

  • Every statement in the proof has a semantics, a meaning, that is invariant (given the remark about variables above).
  • Each statement is justified by some of the previous statements. This justification is given by two systems that the reader is supposed to understand.
  • One system is the rules of symbol manipulation that are applied to the symbolic expressions, ordinary algebra, and higher-level manipulations used in particular branches of math.
  • The other system consists of the rules of logical reasoning that justify the claims that each statement follows logically from preceding ones.
  • These two systems are really branches of one system, the entire system of math computation and reasoning. It can be obscure which system is being used in a particular step.

Suppression of reasons

The logical and symbolic-manipulation reasons justifying the deductions may not be made completely explicit. In fact, for many steps they may not be mentioned at all, and for others, one or two phrases may be used to give a hint. This is standard practice in writing “rigorous” proofs. That is a descriptive statement, made without criticism. Giving all the reasons is essentially impossible without a computer.

I am aware that some work has been done to write proof checkers that can read a theorem like the one we are considering, stated in natural language, and correctly implement the semantics I have described in this list. I don’t know of any references to such work and would appreciate information about it.

Suppression of reasons makes it difficult to mechanically check a proof written in this standard “rigorous” writing style. Basically, you must be at at least the graduate student level to be able to make sense of what is said, and even experienced math research people find it difficult to read a paper in a very different field. Writing the proof so that it can be checked by a proof checker requires understanding of the same sort, and it typically makes the proof much longer.

One hopeful new approach is to write the proofs using homotopy type theory. The pioneers in that field report that the proofs don’t expand nearly as much as is required by first order logic.

Examples of suppression

Here are many examples of suppression in the $\epsilon$-$\delta$ proof above. This is intended to raise your consciousness concerning how nearly opaque writing in math research is to anyone but the cognoscenti.

  • The first sentence of the theorem names $R$ and $f$ and puts constraints on them that can be used to justify statements in the proof. The naming of $R$ and $f$ requires that every occurrence of $R$ in the proof refers to the same mathematical object, and similarly for $f$.

Remark: The savvy reader “knows” the facts stated in (a), possibly entirely subconsciously. For many of us there is no conscious thought of constraints and permanence of naming. My goal is to convince those who teach beginning abstract math course to become conscious of these phenomena. This remark applies to all the following items as well.

  • The second sentence gives $a$ a specific meaning that will be maintained throughout the proof. It also puts constraints on $a$ and an additional constraint on $f$.
  • The third sentence gives a constraint on $R$, $f$ and $a$. It does not give a constraint on $x$, which is a bound variable. Nor does it name $x$ as a specific number with the same meaning in the rest of the proof. (That happens later).
  • The fact that the first three sentences impose constraints on various objects is signaled by the fact that the sentences are introduced by “let” and “suppose”. The savvy reader knows this.
  • The fourth sentence announces that “$f(a)$ is a maximum of $f$ in $R$” is a consequence of the constraints imposed by the preceding three sentences. (In other words, it follows from the context.) This is signaled by the word “then”.
  • The fact that the paragraph is labeled “Theorem” informs us that the fourth sentence is therefore a statement of what is to be proved, and that every constraint imposed by the first three sentences of the Theorem may be used in the proof.
  • In the proof, statements (1), (4), (5) and (6) rewrite the statements in the theorem according to the definitions of the words involved, namely “derivative: “increases”, “decreases” and “maximum”. Rewriting statements according to the definitions of the words involved is a fundamental method for starting a proof.
  • (2) follows from (1) by rewriting using the definition of “limit”. Note that pattern-matching against the definition of limit requires understanding that there is a zero inside the absolute value signs that is not written down. Could a computer proof-checker handle that?
  • (3) follows from (2). The reader or proof-checker must:
    • Know that it is acceptable to put an upper bound on $\delta$ in the definition of limit.
    • Notice that you can move $|x-a|$ out of the denominator because $x\neq a$ by (2).
  • The conclusion in (6) that we much show that $f(x)\lt f(a)$ is now the statement we must prove.

Remark: In the following items, I mention the context of the proof. I am using the word informally here. It is used in some forms of formal logic with a related but more precise meaning. The context consists of the variables you must hold in your head as you read each part of the proof, along with their current constraints. “Current” means the “now” that you are in when considering the step of the proof you are reading right now. I give some references at the end of the post.

  • At the point between (6) and (7), our context consists of $a$, $R$ and $f$ all subject to some constraints. $x$ is not yet in the context of our proof because its previous occurrences in the theorems and in (1) through (6) have been bound, mostly by an unexpressed universal quantifier. Now we are to think of $x$ as a specific number bound by some constraints.
  • The statement in (7) that the case $x\gt a$ as a symmetric proof is a much higher-level claim than the other steps in this proof, even though in fact it is not very high level compared to statements such as “An application of Serre’s spectral sequence shows$\ldots$”. Most mathematicians with even a little experience will read this statement and accept it in the confidence that they will know how to swap “$\lt$” and “$\gt$” in the proof in the correct way (which is a bit picky) to provide a dual proof. Some students might write out the dual proof to make sure they understand it (more likely because writing it out was a class assignment). I await the day that an automated proof checker can handle a statement like this.
  • (8) introduces three new math objects $\epsilon$, $\delta$ and $y$ subject to several constraints. The symbols occur earlier but they are all bound. $\epsilon$ will be fixed in our context from now until (10). The others don’t appear later.
  • (9) consists of several steps of algebraic computation. A cognoscent (I am tired of writing “savvy”) reader first looks at the computation as a whole and notices that it deduces that $|f(x)-f(a)|\lt\epsilon$, which is almost what is to be proved. This helps the reader understand the reason for the calculation. No mention whatever is made in this step of all this stuff that should go through your mind (or the proof-checker’s “mind”).
  • The computations in (9) are are basic algebra not explained step by step, except that the remark that $f(x)\lt f(y)$ explains how you get $f(x)-f(y)+f(y)-f(a) \lt f(y)-f(a)$.
  • (10) banishes $\epsilon$ from the context by universally quantifying over it. That $f(x)\leq f(a)$ follows by the garbage-dump-in-Star-Wars trick that often baffles first year analysis students: Since for all positive $\epsilon$, $f(x)\lt f(a)+\epsilon$, then $f(x)\leq f(a)$. (See also Terry Tao’s article in Tricks Wiki.)
  • (11) “By the same argument as leading up to (10)” puts some demands on the reader, who has to discover that you have to go back to (7) and do the following steps with a new context using a value of $x$ that is halfway closer to $a$ than the “old” $x$ was. This means in particular that the choice of $\frac{x-2}{2}$ is unnecessarily specific. But it works.
  • (12) suppresses the reference to (11).

    • References

    I have written extensively on these topics. Here are some links.

    Rich-rigorous bifurcation in math thinking

The symbolic language

Math English and the language of proofs

Proofs and context

Send to Kindle
exposition, math, proof, representations, understanding math

Modules for mathematical objects

Leave a comment

Notes on viewing.

A recent article in Scientific American mentions discusses the idea that concepts are represented in the brain by clumps of neurons.  Other neuroscientists have proposed that each concept is distributed among millions of neurons, or that each concept corresponds to one neuron.  

I have written many posts about the idea that:  

  • Each mathematical concept is embodied in some kind of module in the brain.
  • This idea is a useful metaphor for understanding how we think about mathematical objects.
  • You don't have to know the details of the method of storage for this metaphor to be useful.  
  • The metaphor clears up a number of paradoxes and conundrums that have agitated philosophers of math.

The SA article inspired me to write about just how such a module may work in some specific cases.  

Integers

Mathematicians normally thinks of a particular integer, say $42$, as some kind of abstract object, and the decimal representation "42" as a representation of the integer, along with XLII and 2A$_{16}$.  You can visualize the physical process like this: 

  • The mathematician has a module Int (clump of neurons or whatever) that represents integers, and a module FT that represents the particular integer $42$. 
  • There is some kind of asymmetric three-way connection from FT to Int and a module EO (for "element of" or "IS_A"). 
  • That the connection is "asymmetric" means that the three modules play different roles in the connection, meaning something like "$42$ IS_A Integer"
  • The connection is a physical connection, not a sentence, and when  FT is alerted ("fired"?), Int and EO are both alerted as well. 
  • That means that if someone asks the mathematician, "Is $42$ an integer?", they answer immediately without having to think about it — it is a random access concept like (for many people) knowing that September has 30 days.
  • The module for $42$ has many other connections to other modules in the brain, and these connections vary among mathematicians.

The preceding description gives no details about how the modules and interconnections are physically processed.  Neuroscientists probably would have lots of ideas about this (with no doubt considerable variation) and would criticize what I wrote as misrepresenting the physical details in some ways.  But the physical details are their job, not mine.  What I claim is that this way of thinking makes it plausible to view abstract objects and their properties and relationships as physical objects in the brain.  You don't have to know the details any more than you have to know the details of how a rainbow works to see it (but you know that a rainbow is a physical phenomenon).

This way of thinking provides a metaphor for thinking about math objects, a metaphor that is plausibly related to what happens in the real world.

Students

A student may have a rather different representation of $42$ in the brain.  For one thing, their module for $42$ may not distinguish the symbol "42" from the number $42$, which is an abstract object.   As a result they ask questions such as, "Is $42$ composite in hexadecimal?"  This phenomenon reveals a complicated situation. 

  • People think they are talking about the same thing when in fact their internal modules for that thing may be very differently connected to other concepts in their brain.
  • Mathematicians generally share many more similarities in their modules for $42$ than people in general do.  When they differ, the differences may be of the sort that one of them is a number theorist, so knows more about $42$ (for example, that it is a Catalan number) than another mathematician does.  Or has read The Hitchhiker's Guide to the Galaxy.
  • Mathematicians also share a stance that there are right and wrong beliefs about mathematical objects, and that there is a received method for distinguishing correct from erroneous statements about a particular kind of object. (I am not saying the method always gives an answer!).
  • Of course, this stance constitutes a module in the brain. 
  • Some philosophers of education believe that this stance is erroneous, that the truth or falsity of statements are merely a matter of social acceptance.
  • In fact, the statements in purple are true of nearly all mathematicians.  
  • The fact that the truth or falsity of statements is merely a matter of social acceptance is also true, but the word "merely" is misleading.
  • The fact is that overwhelming evidence provided by experience shows that the "received method" (proof) for determining the truth of math statements works well and can be depended on. Teachers need to convince their students of this by examples rather that imposing the received method as an authority figure.

Real numbers

A mathematician thinks of a real number as having a decimal representation.

  • The representation is an infinitely long list of decimal digits, together with a location for the decimal point. (Ignoring conventions about infinite strings of zeroes.)
  • There is a metaphor that you can go along the list from left to right and when you do you get a better approximation of the "value" of the real number. (The "value" is typically thought of in terms of the metaphor of a point on the real line.)
  • Mathematicians nevertheless think of the entries in the decimal expansion of a real number as already in existence, even though you may not be able to say what they all are.
  • There is no contradiction between the points of view expressed in the last two bullets.
  • Students frequently do not believe that the decimal entries are "already there".  As a result they may argue fiercely that $.999\ldots$ cannot possibly be the same number as $1$.  (The Wikipedia article on this topic has to be one of the most thoroughly reworked math articles in the encyclopedia.)

All these facts correspond to modules in mathematicians' and students' brains.  There are modules for real number, metaphor, infinite list, decimal digit, decimal expansion, and so on.  This does not mean that the module has a separate link to each one of the digits in the decimal expansion.  The idea that there is an entry at every one of the infinite number of locations is itself a module, and no one has ever discovered a contradiction resulting from holding that belief.

References

  • Brain cells for Grandmother, by Rodrigo Quian Quiroga, Itzhak Fried and Christof Koch.  Scientific American, February 2013, pages 31ff.

Gyre&Gimble posts on modules

Notes on Viewing  

This post uses MathJax. If you see mathematical expressions with dollar signs around them, or badly formatted formulas, try refreshing the screen. Sometimes you have to do it two or three times.

 

 

 
Send to Kindle
exposition, language of math, math, representations, understanding math

Thinking about mathematical objects revisited

3 Comments

How we think about X

It is notable that many questions posted at MathOverflow are like, “How should I think about X?”, where X can be any type of mathematical object (quotient group, scheme, fibration, cohomology and so on).  Some crotchety contributors to that group want the questions to be specific and well-defined, but “how do I think about…” questions  are in my opinion among the most interesting questions on the website.  (See note [a]).

Don’t confuse “How do I think about X” with “What is X really?” (pace Reuben Hersh).  The latter is a philosophical question.  As far as I am concerned, thinking about how to think about X is very important and needs lots of research by mathematicians, educators, and philosophers — for practical reasons: how you think about it helps you do it.   What it really is is no help and anyway no answer may exist.

Inert and eternal

The idea that mathematical objects should be thought of as  “inert” and “eternal”  has been around for awhile.  (Never mind whether they really are inert and eternal.)  I believe, and have said in the past [1], that thinking about them that way clears up a lot of confusion in newbies concerning logical inference.

  • That mathematical objects are “inert” means that the do not cause anything. They have no effect on the real world or on each other.
  • That they are “eternal” means they don’t change over time.

Naturally, a function (a mathematical object) can model change over time, and it can model causation, too, in that it can describe a process that starts in one state and achieves stasis in another state (that is just one way of relation functions to causation).  But when we want to prove something about a type of math object, our metaphorical understanding of them has to lose all its life and color and go dead, like the dry bones before Ezekiel started nagging them.

It’s only mathematical reasoning if it is about dead things

The effect on logical inference can be seen in the fact that “and” is a commutative logical operator. 

  • “x > 1 and x < 3″ means exactly the same thing as “x < 3 and x > 1″
  • “He picked up his umbrella and went outside” does not mean the same thing as “He went outside and picked up his umbrella”.

The most profound effect concerns logical implication.  “If  x > 1 then x > 0″ says nothing to suggest that x > 1 causes it to be the case that x > 0.  It is purely a statement about the inert truth sets of two predicates lying around the mathematical boneyard of objects:  The second set includes the first one.  This makes vacuous implication perfectly obvious.  (The number -1 lies in neither truth set and is irrelevant to the fact of inclusion).

Inert and eternal rethought

There are better metaphors than these.  The point about the number 3 is that you think about it as outside time. In the world where you think about 3 or any other mathematical object, all questions about time are meaningless.

  • In the sentence “3 is a prime”, we need a new tense in English like the tenses ancient (very ancient) Greek and Hebrew were supposed to have (perfect with gnomic meaning), where a fact is asserted without reference to time.
  • Since causation involves this happens, then this happens, all questions about causation are meaningless, too.  It is not true that 3 causes 6 to be composite, while being irrelevant to the fact that 35 is composite.

This single metaphor “outside time” thus can replace the two metaphors “inert” and “eternal” and (I think) shows that the latter two are really two aspects of the same thing.

Caveat

Thinking of math objects as outside time is a Good Thing when you are being rigorous, for example doing a proof.  The colorful, changing, full-of-life way of thinking of math that occurs when you say things like the statements below is vitally necessary for inspiring proofs and for understanding how to apply the mathematics.

  • The harmonic series goes to infinity in a very leisurely fashion.
  • A function is a machine — when you dump in a number it grinds away and spits out another number.
  • At zero, this function vanishes.

Acknowledgment

Thanks to Jody Azzouni for the italics (see [3]).

Notes

a.  Another interesting type of question  “in what setting does such and such a question (or proof) make sense?” .  An example is my question in [2].

References

1.  Proofs without dry bones

2. Where does the generic triangle live?

3. The revolution in technical exposition II.

Send to Kindle
math, proof, representations, understanding math

Proofs without dry bones

3 Comments

I have discussed images, metaphors and proofs in math in two ways:

(A) A mathematical proof

A monk starts at dawn at the bottom of a mountain and goes up a path to the top, arriving there at dusk. The next morning at dawn he begins to go down the path, arriving at dusk at the place he started from on the previous day. Prove that there is a time of day at which he is at the same place on the path on both days.

Proof: Envision both events occurring on the same day, with a monk starting at the top and another starting at the bottom at the same time and doing the same thing the monk did on different days. They are on the same path, so they must meet each other. The time at which they meet is the time required.

This example comes from Fauconnier, Mappings in Thought and Language, Cambridge Univ. Press, 1997. I discuss it in the Handbook, pages 46 and 153. See the Wikipedia article on conceptual blending.

(B) Rigor and rigor mortis

The following is quoted from a previous post here. See also the discussion in abstractmath.

When we are trying to understand or explain math, we may use various kinds of images and metaphors about the subject matter to construct a colorful and rich representation of the mathematical objects and processes involved. I described some of these briefly here. They can involve thinking of abstract things moving and changing and affecting each other.

When we set out to prove some math statement, we go into what I have called “rigorous mode”. We feel that we have to forget some of the color and excitement of the rich view. We must think of math objects as inert and static. They don’t move or change over time and they don’t interact with other objects or the real world. In other words, pretend that all math objects are dead.

We don’t always go all the way into this rigorous mode, but if we use an image or metaphor in a proof and someone challenges us about it, we may rewrite that part to get rid of the colorful representation and replace it by a calculation or line of reasoning that refers to the math objects as if they were inert and static – dead.

I didn’t contradict myself.
I want to clear up some tension between these two ideas.

The argument in (A) is a genuine mathematical proof, just as it is written. It contains hidden assumptions (enthymemes), but all math proofs contain hidden assumptions. My remarks in (B) do not mean that a proof is not a proof until everything goes dead, but that when challenged you have to abandon some of the colorful and kinetic reasoning to make sure you have it right. (This is a standard mathematical technique (note 1).)

One of the hidden assumptions in (A) is that two monks walking the opposite way on the path over the same interval of time will meet each other. This is based on our physical experience. If someone questions this we have several ways to get more rigorous. One many mathematicians might think of is to model the path as a curve in space and consider two different parametrizations by the unit interval that go in opposite directions. This model can then appeal to the intermediate value theorem to assert that there is a point where the two parametrizations give the same value.

I suppose that argument goes all the way to the dead. In the original argument the monk is moving. But the parametrized curve just sits there. The parametrizations are sets of ordered pairs in R x (R x R x R). Nothing is moving. All is dry bones. Ezekiel has not done his thing yet.

This technique works, I think, because it allows classical logic to be correct. It is not correct in everyday life when things are moving and changing and time is passing.

Avoid models; axiomatize directly
But it certainly is not necessary to rigorize this argument by using parametrizations involving the real numbers. You could instead look at the situation of the monk and make some axioms the events being described. For example, you could presumably make axioms on locations on the path that treat the locations as intervals rather than as points.

The idea is to make axioms that state properties that intervals have but doesn’t say they are intervals. For example that there is a relation “higher than” between locations that must be reflexive and transitive but not antisymmetric. I have not done this, but I would propose that you could do this without recreating the classical real numbers by the axioms. (You would presumably be creating the intuitionistic real numbers.)

Of course, we commonly fall into using the real numbers because methods of modeling using real numbers have been worked out in great detail. Why start from scratch?

About the heading on this section: There is a sense in which “axiomatizing directly” is a way of creating a model. Nevertheless there is a distinction between these two approaches, but I am to confused to say anything about this right now.

First order logic.
It is commonly held that if you rigorize a proof enough you could get it all the way down to a proof in first order logic. You could do this in the case of the proof in (A) but there is a genuine problem in doing this that people don’t pay enough attention to.

The point is you replace the path and the monks by mathematical models (a curve in space) and their actions by parametrizations. The resulting argument calls on well known theorems in real analysis and I have no doubt can be turned into a strict first order logic argument. But the resulting argument is no longer about the monk on the path.

The argument in (A) involves our understanding of a possibly real physical situation along with a metaphorical transference in time of the two walks (a transference that takes place in our brain using techniques (conceptual blending) the brain uses every minute of every day). Changing over to using a mathematical model might get something wrong. Even if the argument using parametrized curves doesn’t have any important flaws (and I don’t believe it does) it is still transferring the argument from one situation to another.

Conclusion:
Mathematical arguments are still mathematical arguments whether they refer to mathematical objects or not. A mathematical argument can be challenged and tested by uncovering hidden assumptions and making them explicit as well as by transferring the argument to a classical mathematical situation.

Note 1. Did you ever hear anyone talking about rigor requiring making images and metaphors dead? This is indeed a standard mathematical technique but it is almost always suppressed, or more likely unnoticed. But I am not claiming to be the first one to reveal it to the world. Some of the members of Bourbaki talked this way. (I have lost the reference to this.)

They certainly killed more metaphors than most mathematicians.

Note 2. This discussion about rigor and dead things is itself a metaphor, so it involves a metametaphor. Metaphors always have something misleading about them. Metametaphorical statements have the potential of being far worse. For example, the notion that mathematics contains some kind of absolute truth is the result of bad metametaphorical thinking.

Send to Kindle