"Proving" that definitions "work"

[Mirero]

So I have a friend who wants to become an engineer who is overtly obsessed with his mathematical foundations at the moment. He has confessed recently that he didn't understand the definition of the derivative, and asked me to elaborate. And so I did.

However, what he asked next kind of confused me, and it took me a while to understand what exactly he was asking. He asked me "Why is this definition the way it is", to which I said that it is simply the most convenient way to formulate it. He was still confused, so I tried to explain that definitions simply served as shorthand for concepts that we want to refer to. I emphasized the need for definitions by showing him a non-recursive definition for ordinal arithmetic and showing him how tedious it would be to use that definition without the shorthand $\alpha + \beta$ in everyday life.

He followed that part, but then he asked me to "prove" that this definition "works". I wasn't sure exactly what he means by this, but I assumed that he wanted me to show that the definition was logically consistent. I told him that we derive certain concepts from the axioms, and we give those concepts a name to make it easier to work with them. As long as the deductions are valid, so are our definitions. He doesn't seem to follow that explanation though.

Is there any better way I can explain this to him? Or is it simply a matter of mathematical maturity? His highest math is BC Calculus, getting a 5 on the AP exam, if that is of any relevance.

 

[jedishrfu]

Perhaps you could use engineering terms to describe it like here's a 3" hexagonal bolt. Show him that the bolt has a set of properties that describes and when you see something that matches those properties you can then say it's a bolt.

In math then we make definitions and we make assumptions called axioms like the components and rules of a game and from there we prove theorems based on the definitions and axioms. We can't "prove" a definition or an axiom as they are fundamental to the math at hand.

 

[micromass]

Asking why ordinal addition works is a fair question. It is supposed to rigorize the arithmetic on natural numbers, but this is not really clear from its definition. So you need to think on how to explain that ordinal addition corresponds to our usual (but nonrigorous) concept of addition.

 

[SteamKing]

Or you could toss your friend a copy of Principia Mathematica by Bertrand Russell and A.N. Whitehead (not to be confused with Newton's great opus):

https://en.wikipedia.org/wiki/Principia_Mathematica

I understand that Russell and Whitehead spend quite a few pages defining and proving 1 + 1.

If this isn't enough to convince your friend to keep from asking a lot of awkward questions, then he should switch from engineering to a more philosophical pursuit of knowledge, because engineering, infused as it is with empirical knowledge sprinkled in with the more scientific aspects, will drive your friend completely batty.

 

[homeomorphic]

It's not clear what he means. Did he mean prove that agrees with the way that we normally do addition?

It's not possible to prove that arithmetic is consistent without circular reasoning, by the way. That's essentially what Godel's theorem says.

In the derivative example, that seems a little closer to the kind of question I would ask, which is what is the motivation, but in the case of the derivative, I find the motivation blindingly obvious, so I am not sure what would satisfy him, assuming he's already been through the standard discussions of velocity or zooming in on a function until it looks straight, and secant lines limiting towards the tangent line. If it were me, I could only resort to a cascade of "what are you really asking" questions.

All over the place in math teaching, definitions are given without motivation, like the rule for multiplying matrices. I didn't know what the point of that was for years, but then I studied real analysis, and it came to light that it was defined that way because that is the way to compose the corresponding linear transformations, expressed in terms of the chosen basis. If I had been really on top of things, I might have figured it out in linear algebra much earlier, but I wasn't that on top of things at that point. Very satisfying to find out that I was right that it wasn't just because the book said to do it that way. That probably has a lot to do with why I changed majors from electrical engineering to math, but in the end math turned out to be much, much worse than EE with regard to those sorts of issues because of the extreme, extreme complexity of the theories, such that so much of it had to be taken on faith, in order to get through it fast enough. The only solution to this dilemma is probably to pursue math as a hobbyist.

 

[Mirero]

It's not possible to prove that arithmetic is consistent without circular reasoning, by the way. That's essentially what Godel's theorem says.

Right, oversight on my part, thanks for the catch.

I eventually got him to understand ordinal arithmetic (in a crude way) by taking an example that Kunen used and clarifying upon it.

As for the case with the derivative, I still don't understand what he is asking either. I went over the stereotypical case of Newton and his problem of finding instantaneous velocity, for which he developed a rudimentary form of the derivative. He then asked me he wants to "understand" the difference quotient (Not sure how to convey that fact that its just an expression) and he wants to understand limits (to which he doesn't want to know formalities, not even through a delta epsilon formulation).

I feel like my friend will get his answer eventually as he gets more experienced with math. He is retaking single variable calculus anyway, as he doesn't feel prepared for multivariable, so hopefully his professor can understand his question better than I can.

 

[aikismos]

Right, oversight on my part, thanks for the catch.

I eventually got him to understand ordinal arithmetic (in a crude way) by taking an example that Kunen used and clarifying upon it.

As for the case with the derivative, I still don't understand what he is asking either. I went over the stereotypical case of Newton and his problem of finding instantaneous velocity, for which he developed a rudimentary form of the derivative. He then asked me he wants to "understand" the difference quotient (Not sure how to convey that fact that its just an expression) and he wants to understand limits (to which he doesn't want to know formalities, not even through a delta epsilon formulation).

I feel like my friend will get his answer eventually as he gets more experienced with math. He is retaking single variable calculus anyway, as he doesn't feel prepared for multivariable, so hopefully his professor can understand his question better than I can.

It sounds like your friend is looking for a Royal Road to Geometry, IMO. There are no Royal Roads to Geometry. :D If you have shown him how the slope of secant in functional notation gives rise to taking the limit of a difference quotient and had him manually calculate it on simple polynomials to contrast that with using the rules, and he can do that, but can't understand it, then he has some serious conceptual shortcomings about what math is. That process engages symbolic, numerical, verbal, and graphical thinking, and those are all four domains.

Did you walk your friend through questioning instead of lecturing on it? Did you try to get your friend to draw the conclusions personally? Did you have your friend write out some slopes of secants on a curve with numerical values? Did you try to get him to generalize it using functional notation to see that the DQ is just the definition of a slope? Did you at some point ask your friend to calculate the slope of a line tangent to the curve instead? How about questions to try to provoke thought, like how close to two points have to be for them to become one point, or at what point in calculations does a secant become a tangent? How about your friend's knowledge regarding the divisibility of zero in the rationals? Did you explore your friend's knowledge of the infinite density property of reals and relate it to mapping them to the real number line? Me personally, I believe in the foundations of math arising from the embodied mind, that is, there are certain psychological primitives which give rise to the axioms we choose in math. Have you tried explaining that math comes from the mind? Have you tried to draw analogies between the computer and the mind? Have you tried talking about iteration and infinite regress in simple terms? What about a geometrical and verbal approach by defining the derivative as the slope of the line tangent to a point on the curve? Maybe use Geogebra and do a quick and easy construction with a slider to show the approach of two points on a curve and the graphical and numerical effects on the slope of a line? Another strategy might be to use two graphs, $P(x)$ and $P'(x)$, and have him draw slopes of lines tangent to a point on $P(x)$ by reading $P'(x)$ and vice versa. The same sorts of exercises can be used for ordinal arithmetic, like starting out with vector addition, mapping the operations to planar and volumetric geometry, showing how exponentiation is defined by multiplication by addition by counting...

The foundations of mathematics is something that is NOT intuitive (otherwise Euclid wouldn't have enjoyed his reign for 2,000) years. My experience has been that "understanding" math comes in erratic fits for many students, and is highly individualized to the style and experience of the student.