From: Albert
To: Glyn Holton
Sent: April 12
Subject: Consulting

Hello,

Your expertise and services, as laid out on your web site, are quite impressive. It would be an honor to be associated with a winner such as your company.

I have been a professor of mathematics for some twenty (20) years in a community college. I have two Masters degrees, one in Pure Mathematics and the other in Mathematical Finance, which is equivalent to Quantitative Finance or Financial Engineering.

Given my credentials, I can expertly and confidently teach many of the subjects (especially the mathematical ones) in your curriculum. I wonder if I can be of use to your organization. I’d be glad to email you a copy of my resume. Please feel free to let me know what you think about my proposal.

Thank you.

Sincerely,

Albert

From: Glyn Holton
To: Albert
Sent: April 15
Subject: Re: Consulting

The symbol dy/dx … does it represent a fraction or a whole symbol?

From: Albert
To: Glyn Holton
Sent: April 15
Subject: Re: Consulting

Hello,

I just opened your email.

The symbol dy/dx represents the derivative of (the function) y with respect to x. So, the symbol dy/dx represents a single idea—the idea of the derivative. Therefore it is a whole symbol. This symbol was invented by Leibniz, the other contemporaneous inventor of the Calculus.

I could say more about your question. But I will leave it at that at this point.

Please feel free to ask further questions.

Sincerely,

Albert

From: Glyn Holton
To: Albert
Sent: April 15
Subject: Re: Consulting

Hi Albert: I accept your history but not your math. Why do you reject dy/dx as a fraction — a ratio of differentials? You said you could say more on the topic … Cheers, Glyn

From: Albert
To: Glyn Holton
Sent: April 15
Subject: Re: Consulting

Hi,

That’s my point! I did say I could say more.

You can treat them as (infinitesimal) differentials. But there is and has been a whole lot of philosophical/logical debate as to what an infinitesimal actually is. This area of discussion – as exciting, rigorous and important as it is – is beyond my expertise.

In fact, when it comes to practicality, we treat dy/dx as if it were a fraction. As mentioned, this happens, for example, when we want to find the differential of a function f(x) as in dy=f’(x)dx. Or, sometimes we write dy/dx=1/(dx/dy) or in the chain rule case: dy/dx = (dy/du) (du/dx).

Anyway, getting back to the point, the symbol dy/dx represents – MORE THAN ANYTHING ELSE – the limit of ∆y/∆x, where ∆y = f(x+∆x) – f(x), as ∆x approaches zero, if the limit exists. That limit is singular. Now, granted ∆y/∆x is indeed a fraction as long as ∆x≠0, which latter condition is the case at all times, anyway. If we treat dy and dx as infinitesimal quantities such that dx≠0, and yet exceedingly close to zero, then we may treat dy/dx as a fraction.

Sorry, if I have used too many words to express my thoughts. But, that’s my response.

It’s been fun exchanging a bit of math with you. Please keep communicating.

If you are still interested in receiving a copy of my resume, please let me know.

Sincerely,

Albert

From: Glyn Holton
To: Albert
Sent: April 15
Subject: Re: Consulting

What is an infinitesimal? What is a differential? – Glyn

From: Albert
To: Glyn Holton
Sent: April 15
Subject: Re: Consulting

Hi Glyn,

Where do I begin?

As I noted earlier, the philosophical/logical questions surrounding the infinitesimal are, basically, beyond my expertise. There is quite a bit (or a lot) of history behind it. It was introduced by the ancient Greeks. In the early stages of the development of the Calculus the infinitesimal was considered necessary and important. But Weierstrass made obsolete the use of infinitesimals in mathematics.

A definition, albeit unsatisfactory, of infinitesimal is that it represents a positive quantity that is less than any other positive quantity. In the system of real numbers, no such thing could exist.

Notwithstanding the preceding, a colleague of mine has claimed that it is possible to give the notion of the infinitesimal a non-contradictory treatment in a certain logico-deductive system. Once I sat in a lecture attempting to do just this.

Given a function f(x) of a single variable x, the differential dy is defined as dy=f’(x)dx. But then the question is: what does dx represent? Yes, dx is the differential of x itself. But what is it? Some have defined it as a quantity that approaches zero but never becomes zero. But, demanding rigor, this creates more headache in math as the word “approaches” creates a new mystery. But it does give an intuitive sense as to what is happening.

One way to define dy is to consider it as a function of two variables, let’s say, x and h, that has the following property:

dy(x,h) = f’(x) h + q(x,h) for some function q such that limit of q(x,h)/h is zero as h approaches zero. Clearly this does coincide with dy=f’(x)dx (where dx=h) as dx approaches (but never becomes) zero.

Now, the idea of differential is of course applicable to multi-variable functions. Without getting into too much symbolism here, I’d like to mention that author Walter Rudin in his classic book REAL AND COMPLEX ANALYSIS (ISBN 0070542341), third edition, on page 150 states “The term differential is also often used for T’(x)” where T is a function of x. Earlier he calls T’(x) as the derivative of T at x (where x may be thought of as a vector here).

Another way, yet, to verbalize the idea of the differential is to say that it is the rate at which the function f(x) changes with respect to x. The slightest change dx in x results in a change dy in y=f(x) where dy=f’(x)dx.

Fortunately, when it comes to doing math itself, we hardly need to spend time on the hairsplitting aspects of the dx or dy or the infinitesimal. We just treat dx and dy formally, and move on from there.

I feel I have accomplished very little in terms of explicating the question of what exactly an infinitesimal is. Perhaps we need not be concerned about the philosophical/logical aspects of the infinitesimal. We need to make sure, though, that its FORMAL use in math does not lead to contradictions. So far, so good.

But, instead of spending time on the fine details of dx or dy as I have done so far, why not testing me with some math questions? But really, what a teacher should be concerned about is not only knowledge of the subject matter, but the delivery of the subject, just as much. So, if you like, you may either ask me more questions, of any sort you like, or ask me to make a presentation of a mathematical idea, or something like that.

Feel free to ask more questions.

Albert

From: Albert
To: Glyn Holton
Sent: April 23
Subject: Fw: Consulting

Dear Mr. Glyn Holton,

Wondering if our email exchange has run its course. Has it?

Please feel free to express your perception/evaluation of my math knowledge, etc. I have a thick skin and can take it. Please go for it unabashedly!

At any rate, would like to thank you for the math challenges you put before me. It was a unique experience.

Sincerely,

Albert

**** I will not send you emails unless you send one first.****

From: Glyn Holton
To: Albert
Sent: April 24
Subject: Re: Consulting

Hi Albert:

Sorry, I just got distracted.

Do an experiment. Get a standard college calculus text and look up “infinitesimal” in the index. I have tried with about ten calculus texts myself and never found a single mention of infinitesimals. During the 1800s, mathematicians, and especially Cauchy, finally got around to rigorizing calculus. They got rid of the “infinitesimal” business once and for all, replacing infinitesimals with limits. It is troubling how widespread misunderstanding of calculus is 150 years later. Instead of understanding calculus from Cauchy’s rigorous standpoint, people embrace a hodge-podge of infinitesimals AND limits. Thousands of books, many in finance, treat infinitesimals as if they were rigorous—and claim that expressions using differentials are informal. It is no wonder that people are so profoundly confused when they go on to try to learn stochastic calculus.

Anyway, my three courses, Math 1, Math 2 and Math 3, are my way of addressing the problem, at least in finance circles.

I’m not hiring, but maybe I have helped you another way.

Kind Regards,

Glyn

From: Albert
To: Glyn Holton
Sent: April 26
Subject: Infinitesimals and Non-Standard Analysis

Hi Glyn,

You are correct about the standard calculus texts never making a single mention of infinitesimals, which was my conclusion after some examination of my own collection of math books, running at several hundred, not to mention the internet.

But the question you pressed on was what an infinitesimal is. I did do a little more research, and sure enough the idea of infinitesimal can be and has been given a rigorous treatment in non-standard analysis. One mathematician who has done significant work in this area is Abraham Robinson. A search of Wikipedia reveals quite a bit about him. If interested, you may link to http://en.wikipedia.org/wiki/Abraham_Robinson for further info. Excerpts of the book Non-Standard Analysis by Abraham Robinson can be seen here. So, there was a historic period when getting rid of the infinitesimal, as understood at that time, was the right thing to do. But, Abraham Robinson, about whose work I know nothing, cannot be ignored off-hand.

So, I ask myself: why do I bother to mention the word ‘infinitesimal’ in my classes? It invariably happens when my students demand to know at an intuitive level what dx is. As we have seen, many books so often equate dx with ∆x when ∆x is very small, which is, strictly speaking, incorrect, but intuitively helpful, or at least seemingly helpful.

Thanks for the informative exchanges. Good luck with your endeavors.

Sincerely,

Albert

From: Glyn Holton
To: Albert
Sent: April 26
Subject: Re: Infinitesimals and Non-Standard Analysis

Hi Albert:

Non-standard analysis is what happened when a cheeky mathematician, Abraham Robinson, thought it would be clever to reinsert infinitesimals in mathematics. He quite formally did so, extending the real numbers to include new constructs called infinitesimals—much as mathematicians two centuries earlier extended the real numbers to include imaginary numbers. Robinson defined his infinitesimal constructs in such a way that he could then formally reproduce results in calculus using them.

Problems with non-standard analysis are that

1. It is far more difficult to learn than calculus. If we ever embraced it, students would have to first get a Ph.D. in math before they could learn elementary calculus. This makes non-standard analysis as a means of making calculus more intuitive akin to throwing the baby out with the bathwater.

2. It is unnecessary, since calculus is perfectly fine based on limits.

3. It has produced no meaningful applications other than to allow frustrated students to claim that infinitesimals DO EXIST.

You say

“As we have seen, many books so often equate dx with ∆x when ∆x is very small, which is, strictly speaking, incorrect, but intuitively helpful, or at least seemingly helpful.”

Why is it incorrect to set dx = ∆x? Can you give me a rigorous answer … one that doesn’t rely on infinitesimals (of either the “hand waving” or the “non-standard analysis” form)?

Glyn

From: Albert
To: Glyn Holton
Sent: April 26
Subject: Re: Infinitesimals and Non-Standard Analysis

Hi Glyn,

You asked: ” Why is it incorrect to set dx = ∆x?”

Here’s what I think: No matter how small ∆x is, it is NOT small enough! So, strictly speaking, it will never do. Granted, this answer so far is not rigorous.

The problem with the content of our exchanges — at least those coming from my side — is that we have mixed some rigor with a lot of non-rigorous speech. With this in the backdrop, I will try to answer your question somewhat more rigorously: Given the fact that the symbol dx remains undefined up to this point, the question, such as it is, cannot be answered.

A better approach is that of using the idea of linearization of the function.

Simply, define the function df as df: RxR —> R such that df(x,h)=f'(x)h.

Now, we don’t need to bother with infinitesimal, and what not.

I am curious to know as to how you define dx.

Albert

From: Glyn Holton
To: Albert
Sent: April 26
Subject: Re: Infinitesimals and Non-Standard Analysis

Hi Albert:

I like this:

> Simply, define the function df as df: RxR —> R such that df(x,h)=f'(x)h.

It is rigorous, but you are unsure of yourself, which is why you introduced h. Should h represent dx, ∆x or something else? Also, do you intend that df(x,h) represent the more familiar dy, or do you see these as different?

I will be happy to define dx for you, but let me give you another chance to do it yourself. Here is a hint. Try defining ∆x first.

Cheers,

Glyn

From: Albert
To: Glyn Holton
Sent: April 26
Subject: Re: Infinitesimals and Non-Standard Analysis

Hi Glyn,

No, I am not unsure of myself. It is only a matter of symbolic expediency that I used h instead of dx. The symbol h is just a variable, just like dx is. Again, not wanting to get into detailed discussion, you may ‘safely’ let df and dy represent the same thing.

Albert

From: Glyn Holton
To: Albert
Sent: April 26
Subject: Re: Infinitesimals and Non-Standard Analysis

Fair enough: h = dx. You are very close to giving me a rigorous definition of ∆x and dx. Can you do it, or do you want me to? – Glyn

From: Albert
To: Glyn Holton
Sent: April 26
Subject: Re: Infinitesimals and Non-Standard Analysis

Hi Glyn,

Go right ahead!

Albert

From: Glyn Holton
To: Albert
Sent: April 29
Subject: RE: Infinitesimals and Non-Standard Analysis

Hi Albert:

Go back to your linearization (i.e. slope of a tangent line) formula.

df(x,h)=f'(x)h

This is linear if we consider x at a fixed point and let h vary. That means that we are treating x as a constant, h as an independent variable and df as a dependent variable—h can take on any real value, and so can df.

In recent e-mails, you agreed to certain changes in notation, specifically dy = df and dx = h. People who are still struggling with the whole infinitesimal business often get a little nervous when I make a substitution like this because they associate the terms dy and dx with infinitesimals. They wonder “can we do this?” Just forget you have ever heard of dy, dx or infinitesimals before. I am DEFINING dx = h, and your linearization formula becomes my definition for dy:

dy = f'(x)dx

Now, f’(x) is well defined by the definition of the derivative, and dx is well defined as an independent variable, so dy is well defined. Since dx and dy are just plain old independent and dependent variables that can take on any real values, I can divide both sides by dx to obtain

dy/dx = f'(x)

For a fixed x, f'(x) is a constant and we see that, although dx and dy are variables, their ratio is a constant—no surprise. This is what your linearization is about.

If you are still uncomfortable with this, let’s try a different approach using ∆x and ∆y. As with our discussion of dx and dy, consider a fixed point x and let ∆x be an independent variable. Then ∆y is a dependent variable defined by

∆y = f(x + ∆x) – f(x)

Again, all terms are well defined, so ∆y is well defined. Our formula is now non-linear, and ∆y represents the exact change in the function f over the interval (x, x + ∆x).

Compare this to the earlier formula

dy = f'(x)dx

that defines dy as a linear approximation for the change in the function f over the interval (x, x + dx).

Here is an example. Let f(x) = x^2 and x = 5. The terms ∆x and dx are independent variables, so we can set them equal to whatever we like. Let’s say ∆x = dx = 3. Then

∆y = f(x + ∆x) – f(x) = f(8) – f(5) = 39

is the exact change in f over the interval (5,8) while

dy = f'(x)dx = (10)(3) = 30

is a linear approximation for the change in f over the same interval.

Here is a slide from my Financial Math 1 course.

The graph on the right is called Barrow’s Triangle after Isaac Barrow (1630-1677). I have depicted it using Leibniz’s notation, but Barrow was a predecessor of both Newton and Leibniz. Here is something that may surprise you: Barrow was Newton’s teacher and he also instructed Leibniz in his mathematical research shortly before Leibniz developed his own calculus.

In Barrow’s triangle, we set ∆x = dx. We can do this because they are independent variables. We see that ∆y indicates the exact change in y over the interval (x, x + ∆x) while dy indicates a linear approximation of the change in y over that same interval.

There you have it: a rigorous definition of dx and dy that doesn’t involve infinitesimals. It doesn’t even require that dy or dx be “small.” Furthermore, it is profoundly more simple (i.e. intuitive) than anything related to infinitesimals.

If you make Barrow’s triangle your starting point, stochastic calculus is easy to learn. This is important because there are thousands of quantitative professionals who struggle with stochastic calculus. The problem is that, confused by infinitesimals, they don’t really understand calculus. If you really understand the basics—calculus and probability—stochastic calculus is easy.

Kind Regards,

Glyn