David R. Henderson  

Rigor, Math, and Numbers

Does High School Algebra Pass ... As Goes Janesville ...

Quickly researching the work of the two Nobel Prize winners Monday morning has given me more than the usual amount of thinking to blog on. I came across an interesting thought in the classic 1962 Gale/Shapley article that, as you'll see, it did not make sense to put in my article in the Wall Street Journal. But it was insightful and well-stated. It reminded me of why I had become a math major at the University of Winnipeg, although I admit that my math skills (though not my algebra skills, Bryan) have atrophied.

Here's the short verbal but mathematical proof from David Gale and Lloyd Shapley that led to the insightful quote to come:

We assert that this set of marriages is stable. Namely, suppose John and Mary are not married to each other but John prefers Mary to his own wife. Then John must have proposed to Mary at some stage and subsequently been rejected in favor of someone that Mary liked better. It is now clear that Mary must prefer her husband to John and there is no instability.

Now the quote about math:
Finally, we call attention to one additional aspect of the preceding analysis which may be of interest to teachers of mathematics. This is the fact that our result provides a handy counterexample to some of the stereotypes which non-mathematicians believe mathematics to be concerned with.

Most mathematicians at one time or another have probably found themselves in the position of trying to refute the notion that they are people with "a head for figures." or that they "know a lot of formulas." At such times it may be convenient to have an illustration at hand to show that mathematics need not be concerned with figures, either numerical or geometrical. For this purpose we recommend the statement and proof of our Theorem 1. The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical, while people without mathematical training will probably find difficulty in following the argument, though not because of unfamiliarity with the subject matter.

What, then, to raise the old question once more, is mathematics? The answer, it appears, is that any argument which is carried out with sufficient precision is mathematical, and the reason that your friends and ours cannot understand mathematics is not because they have no head for figures, but because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences. This observation will hardly be news to those engaged in the teaching of mathematics, but it may not be so readily accepted by people outside of the profession. For them the foregoing may serve as a useful illustration.

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CATEGORIES: Economic Methods

COMMENTS (13 to date)
Ken B writes:

This matches my experience. I am always struck, as a trained mathematician, by how many people take my reductio ad absurdum as proof of the absurdity, and object.

...because they are unable [or unwilling, DRH] to achieve the degree of concentration required to follow a moderately involved sequence of inferences.

Of which both Romney and Obama seem to be aware, judging by last night's debate.

BZ writes:

Huh .. so if formal logic is "mathematics", why is it not taught in math classes, but in philosophy classes? I mean, if mathematics is really just about logic and not the specific logic of countable distinctions (iow, numbers), then they did a fantastic job of hiding it when I was in college! My studies of logic did not delve specifically into countable things, and my math classes did not discuss qualitative inferences.

BZ writes:

Doh -- I should have said "informal" logic above, instead of formal. I did take a course called "discrete math" that covered logic tables, symbolic logic, and other formal mechanisms.

Alex Godofsky writes:

BZ: I guess you just had a pretty limited curriculum then? Plenty of my classes made exactly the sort of argument as in the OP; in fact, one of those math classes was where I first encountered that specific proof.

Steve Sailer writes:

"that mathematics need not be concerned with figures, either numerical or geometrical"

So, John likes Mary's figure?

another bob writes:

"there is no instability"

has either of these two nerds ever dated a woman?

Ken Plahn writes:

Aggregating mating preference so that it can be represented in one dimension? That's too Keynesian for me. but it might work...

I could throw in some stimulus to overcome my shortcomings and increase her aggregate demand (preference) for me. That would result in a boom, but alas it would be unsustainable and followed by a bust cycle. I was so close! I should have used more stimulus (bigger diamond).

I wrote a blog for Men's Psychology on this topic.

JGW writes:

I think the discussion surrounding the definition of mathematics is a very interesting one and parallels some discussions on the econlib forum quite remarkably.

One interpretation of the claim is that it is stating mathematics as being a mode of argument, ultimately a mode of reasoning and a mode thinking in general. A conceptual framework or vehicle for understanding. And a very precise one at that.

This is the claim and there is reason as to why there would be frustration with a false perception that it is solely figures and formulas.

Now compare a recurring claim on Russ' podcast:

Most economists are likely confronted with a situation where they are told that they are solely concerned with money and markets, usually just the stock market. This, however, is wrongfully narrow. Economics and its beauty need nothing more than words. It is a field about analyzing interactions and trade offs and ultimately understanding how things work. What is economics? It is similarly argued as a frame of thinking and vehicle for understanding.

The train doesn't stop with these two fields. Similar comparisons and arguments can be made for the likes of philosophy, psychology, chemistry, biology, history, and many more. Anyone in a field and passionate about it is likely to get frustrated with stereotyped perceptions of the field and they would argue that the true definition is something much more conceptual and complex that ultimately yields understanding.

It's a beautiful thing. It's arguably academia flourishing.

I am particularly intrigued by the comparisons that can be made across all fields (and sure the comparisons may become a bit more of a stretch in certain areas).

I guess my ultimate reaction to this piece is that the claim of a field being more conceptual and powerful than it is commonly understood is definitely valid. At the end of the day, though, math is numbers, economics is money, chemistry is explosions and labs, psychoanalysis is lying on a bed blabbering, history is boing, and politics is for the dogs.

Fascinating conversation regardless. Thanks for the article.

Joe Cushing writes:

Recently, in YouTube comments, a person made the claim that the constitution is great and that the problem is that the people of government don't follow it. I pointed out that the people not following the constitution is a flaw of the constitution. No amount of explanation or logic could bring this point home. We went back and forth for several comments. What seemed so obvious to me, I was unable to explain to someone else. Now I haven't even tried to explain it here but I suspect the readers here will understand my logic without explanation. Am I right? I never thought of it as math but it is the same kind of logic.

Shane L writes:

I'm reminded of The Autobiography of Malcolm X, oddly enough, where I think X described a West Indian man called Archie who used his uncanny memory and mathematical skills in gambling rackets. X saw this as hugely wasted potential, and told Archie:

"...that his brain, which could tape-record hundreds of number combinations a day, should have been put at the service of mathematics or science."
Slightly different from David's point here, but perhaps another example of someone who is involved in maths without realising it. When I was in school I knew boys who were terrible at maths, or at least unmotivated, yet were fascinated by and skillful at the application of logic to motor engineering or architecture.
Daniel Artz writes:

While I am not a mathematician, just a lawyer who enjoys mathematics, I do agree that far too many people assume that math is just about numbers. My usual response is to point out that there are a great many areas of math in which numbers play at most a minor role, if any. Saying that math is about numbers is akin to saying that literature is about words. But I don't like the fuzzy definition offered here - that math is simply about a precise mode of thinking. I was taught fairly early, by a very good high school math teacher, that mathematics was the science of patterns - recognizing, understanding, and manipulating patterns. I liked that definition, and it really helped me to understand and appreciate the rigor of mathematical proofs. I fear that a great part of our problems with basic mathematics teaching today is that far too few teachers know and appreciate what math is truly about - they teach math as a set of rules without any unifying principle.

Ken B writes:

Here, for those seeking practice, is a related puzzle.

My wife and I went to a dinner party with four other couples. At the beginning of the party, everyone shook hands with the people they just met. During the party I surveyed all the other people as to how many hands each one shook. I got different answers from each.

So, how many hands did my wife shake?

I hope, if you've never seen this before, that the question shocked you. If it did then you are feeling the pull of mathematics. The solution uses similar reasoning to the proof DRH quotes.
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