### What's wrong with this sentence?

Megan has been pestering me to let her start calculus (precocious soon-to-be-seventh grader that she is). So I brought my old college textbook home from the office, and began reading the introduction.

Here's the first sentence from the textbook's introduction: "Calculus is the mathematical tool used to analyze changes in physical quantities."

What is false in that sentence?

For extra credit, provide a specific counterexample and solve it.

Here's the first sentence from the textbook's introduction: "Calculus is the mathematical tool used to analyze changes in physical quantities."

What is false in that sentence?

For extra credit, provide a specific counterexample and solve it.

## 3 Comments:

I have no idea, but I hope you'll post the answer at some point.

I was never into math. Did all right in algrebra, geometry was hard since I couldn't remember any of the logorhythms (hmmm is that the right term - it's been so long).

Actually, I'm interested in studying math again. My son is terrbile at it, but I can't help him. I tried to find some decent tutorials online, but I really have no idea how to find something good. I mean, the stuff I would find online so I could use it would have to be English. And while my son's 2nd language is English, his 1st and school language is Japanese. It is becoming more frustrating!

Anyway, I'll be on the lookout for your answer!

I plan to post the answer in the comments in a day or so. I would wager that the textbook itself contains counterexamples.

What's wrong in the sentence is it's limitation of calculus to "physical" quantities.

Economists, political scientists, and sociologists who use mathematical models to represent preferences -- called utility functions -- quite often use calculus to characterize properties of those functions and the behavior those models predict.

These functions do not represent "physical quantities" at all.

I would wager that the textbook itself includes some of the canonical economic problems that use calculus (e.g., constrained maximization problems).

Also, statistics uses a lot of calculus, and it is unhinged from "physical quantities." Many, if not most, statisticians would argue that probabilities are subjective imputations (at least I guess the Bayesians would).

So calculus techniques are used by many, many students and scholars who do not study physical quantities with it.

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